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Composites Part B

journal homepage: www.elsevier.com/locate/compositesb

A cost-effective isogeometric approach for composite plates based on a stressrecovery procedure

John-Eric Dufoura,∗, Pablo Antolinb, Giancarlo Sangallic,d, Ferdinando Auricchioa,d,Alessandro Realia,d,e

a Department of Civil Engineering and Architecture, University of Pavia, Via Ferrata, 3, 27100 Pavia, Italyb Institute of Mathematics, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerlandc Department of Mathematics, University of Pavia, Via Ferrata 5, 27100 Pavia, Italyd Instituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” (CNR), Italye Technische Universität München, Institute for Advanced Study, Lichtenbergstraße 2a, 85748 Garching, Germany

A R T I C L E I N F O

Keywords:Isogeometric analysisLaminated composite plateStress recovery procedurePost-processing

A B S T R A C T

This paper introduces a cost-effective strategy to simulate the behavior of laminated plates by means of iso-geometric 3D solid elements. Exploiting the high continuity of spline functions and their properties, a proper out-of-plane stress state is recovered from a coarse displacement solution using a post-processing step based on theenforcement of equilibrium in strong form. Appealing results are obtained and the method is shown to beparticularly effective on slender composite stacks with a large number of layers. These are indeed the caseswhere traditional (e.g., “layerwise”) approaches are more computationally heavy and where researchers aremore inclined to look for alternatives, making the proposed method a very attractive solution.

1. Introduction

Composite materials are used in a wide variety of fields such asaerospace or automotive. The study of composite laminates has becomemore and more important along the past years especially because oftheir light weight and very resistant mechanical properties. A compo-site laminate is usually made of several layers of highly resistant fibersembedded in soft matrix. Laminate structures tend to be prone to da-mage at the interfaces between layers, this mode of failure being re-ferred to as delamination. The prediction and evaluation of damage incomposite laminates demands an accurate evaluation of the three-di-mensional stress state through the thickness, although most of thestudies available in the literature consider the laminate as a two-di-mensional object. Classical two-dimensional theories such as shell ap-proaches are not accurate enough to reliably predict interlaminar da-mage and delamination. Accordingly, a number of layerwise theorieshave been developed to compute more accurately the mechanical stateinside the laminate. Such methods often rely on heavy computationsand hybrid approaches in order to be able to capture the complex be-havior of the interlaminar interfaces. The numerical counterpart of suchlayerwise or hybrid theories was approached mostly using standardfinite element discretization (see, e.g. [1,2], and references therein).

Over the last decade, many novel methods have been proposed to

enhance the standard finite element framework. Among them,Isogeometric Analysis (IGA) is a concept proposed by Hughes et al. [3]where the shape functions used in computer aided design are also ap-proximating physical fields and state variables. It thus relies on splinefunctions, like, e.g., NURBS (Non Uniform Rational B-Splines), to ap-proximate both the geometry and the solution. In addition to improvethe pipeline between design and analysis, spline shape functions possessproperties which can be used to drastically improve the performance ofnumerical analysis compared to standard finite element method. Forexample, spline shape functions can provide easily high order approx-imation and highly simplify the refinement process. Moreover, theirsmoothness guarantees higher accuracy and opens the door to thediscretization of high-order PDEs in primal form. IGA has been suc-cessfully used to tackle a large variety of problems, including solid andstructures (see, e.g. [4–8]), fluids (see, e.g. [9,10]), fluid-structure in-teraction (see, e.g. [11,12]), and other fields (see e.g. [13,14]).

IGA methods have already been used to solve composite laminateproblems. A wide variety of laminated models have been implementedand studied within the IGA framework, especially relying on high-ordertheories [15–18]. These techniques mostly utilize enhanced shell andplate theories. In addition to these 2D approaches, some methods canbe used to compute the full 3D stress state using 3D isogeometricanalysis such as in Refs. [18–20]. In such approaches, each ply of the

https://doi.org/10.1016/j.compositesb.2017.11.026Received 16 June 2017; Received in revised form 11 October 2017; Accepted 15 November 2017

∗ Corresponding author.E-mail address: johnericpierre.dufour@unipv.it (J.-E. Dufour).

Composites Part B 138 (2018) 12–18

Available online 21 November 20171359-8368/ © 2017 Elsevier Ltd. All rights reserved.

T

laminate is delimited by a C0-continuous interface. Such an approach(referred in the following as “layerwise”) involves a large number ofdegrees of freedom when a lot of layers are composing the laminate.Thus it may be not completely satisfactory, despite its accuracy, due toits high cost.

This paper presents an approach consisting in using 3D computa-tions with a reduced number (namely, one) of elements through thethickness and a layerwise integration rule, which is then post-processedin order to obtain an accurate 3D stress state. The post-processing relieson the integration of the equilibrium equations (such as in the recoverytechniques proposed in Refs. [16,21–26]) to compute the stressesthrough the thickness from the in-plane ones. This allows to drasticallyreduce the computational time compared to the 3D layerwise approachas the number of degrees of freedom is significantly decreased. Thepost-processing operation is in fact very fast and its cost does not in-crease significantly with the number of degrees of freedom. Solutionsobtained using the proposed technique are actually close to thoseprovided by full “layerwise” 3D isogeometric analysis, at a fraction ofthe cost, in particular when many layers are present (which is indeedthe case where layerwise approaches are extremely demanding in termsof computational cost, while accurate and inexpensive alternativestrategies are desirable).

The effectiveness of the proposed approach relies on two in-gredients, both granted by the peculiar properties of IGA, namely:

(i) The capability of obtaining accurate in-plane results with a coarsemesh with only one element through the thickness;

(ii) The higher continuity allowing an accurate computation of stressesand of their derivatives from displacements.

The structure of the paper is as follows. First, the considered iso-geometric strategies are briefly introduced. Then, the 3D geometry andthe mechanical problem used as a test case is detailed along with itsanalytical solution. Results using the two considered standard isogeo-metric approaches are then presented. Finally, the post-processing ap-proach is proposed along with numerical results and a comparison withthose provided by full “layerwise” 3D isogeometric analysis. Severalcases using different numbers of quadrature points per material layer,numbers of layers and thicknesses are considered to show the effec-tiveness of the method.

2. Standard IGA strategies for 3D analysis of laminates

In this section we present a preliminary discussion on standard IGAstrategies for 3D analysis of laminates. For the sake of conciseness weavoid reporting an introduction on the basic concepts of IGA for 3Delasticity, including basics of B-Splines, NURBS, etc. Readers are insteadrefered to [27] and references herein. Here, we just recall that we willconsider a standard IGA consisting of an isoparametric Galerkin for-mulation where splines are used for approximating both geometry anddisplacements. As opposed to the framework typically used to computelaminate shell and plate solutions [1,16,28], we choose here to performa full “layerwise” 3D computation. Such an approach has already beenused in combination with IGA in order to solve multilayered plateproblems [18–20] and along those lines we start considering two dif-ferent sets of shape functions, as follows:

2.1. Layerwise approach

The first considered approach is similar to high order finite elementmethods. Each layer is in fact modeled by one patch through thethickness and C0 continuity is kept between elements (see Fig. 1(a)).Such an approach is completely layerwise and, as a consequence, thenumber of degrees of freedom is proportional to the number of layers. Astandard integration rule is adopted, namely p+1 points in each di-rection (p being the degree of the shape functions). This approach is

equivalent to the one proposed in Refs. [18,20].

2.2. Single-element approach

The second approach actually uses a single element through thethickness (see Fig. 1(b)), strongly reducing the number of degrees offreedom with respect to the previously mentioned method. To accountfor the presence of the layers, a special integration rule is adopted,consisting of a q-point Gauss rule ( ≥q 1) over each layer. This can beconsidered as a special homogenized approach with a layerwise in-tegration. Such a method would a priori not give sufficiently good re-sults (in terms of through-the-thickness stress description) but could beeasily coupled with a post-processor to improve the solution, and thiswill be the object of Section 4.

It should be noted that the in-plane continuity of the shape functionsin both approaches is the same.

3. Test case: the Pagano layered plate

The test case used in this study is the classical one proposed byPagano [29]. It consists of a simply supported multilayered 3D platewith a sinusoidal loading on top and a loading-free bottom face (seeFig. 2). This problem can be easily parameterized which allows toanalyze many cases in terms of layer number and distribution. In thefollowing study, a few examples are considered using different numbersof layers (i.e., 3, 4, 11, and 34). In all these cases, the loading conditionsare the same (namely, a two dimensional sinus with a period equal totwice the length of the plate), while the thickness of every single layer isset to 1 mm, and the length of the plate is chosen to be S times largerthan the total thickness t of the laminate. Fig. 2 shows the summary ofthe test problem in the case of three layers.

The laminate is composed by orthotropic layers placed orthogonallyon top of each other (thus creating a 90/0/90/ … laminate). For eachlayer, we have:

=σ ε (1)

where the elasticity tensor can be expressed, using Voigt notation, as:

=

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

− −

− −

− −

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

,

EνE

νE

νE E

νE

νE

νE E

G

G

G

1

1

1

1

1

1

1

122

131

212 2

232

313

323 3

23

31

12

(2)

and the material parameters for these layers are the following:

= =

= == = =

E E

G Gν ν ν 0.25

E

G

2 3 25

12 13 2.5

12 13 23

1

23

(3)

The pressure field used as the top boundary condition is

= ⎛⎝

⎞⎠

⎛⎝

⎞⎠

p x y σ πxSt

πySt

( , ) sin sin0 (4)

In our test, we always select =E 251 GPa, =G 0.523 GPa and =σ 10 .

3.1. Analytical results

Each stress component presents a different distribution alongthickness. In particular, in-plane normal components of the stress tensor(i.e., σ11 and σ22) are discontinuous along the thickness although thedisplacement solution is continuous in all cases. This is obviously due tothe differences in term of material parameters between the layers. On

J.-E. Dufour et al. Composites Part B 138 (2018) 12–18

13

the other hand, equilibrium implies that through-the-thickness stresscomponents have to be C0-continuous (i.e., σ13, σ23 and σ33). Along thearticle, stress and displacement components are normalized, accordingto [29], using the following formulas:

= = =

= =

=

=

σ i j

σ i

σ

u

1,2 1,2

1,2

ijσ

σ S

iσσ SσσE u

σ tS

3

33

10

ij

i0 2

30330

30 3 (5)

Figs. 3 and 4 report the analytical solution (computed using theequations given in Ref. [29]) for all the components of the stress tensorin the case of 3 and 11 layers (blue solid lines). We can clearly identifythe different behavior of the layers, with a “soft” and a “hard” direction.

3.2. First results

In this section, we comment the results obtained using the layerwiseand the single-element approaches, which are compared with theanalytical solutions in Figs. 3 and 4 for the cases of 3 and 11 layers atthe position =X L0.25 and =Y L0.25 . Results are given for an in-planemesh composed of ×9 9 quartic elements, with cubic shape functionsthrough the thickness. All the results shown in this work have beenobtained using an in-house code based on the igatools library (see Ref.[30] for further details).

The layerwise approach captures accurately the solution for bothdisplacements and stresses, no matter if they are discontinuous or not(see Figs. 3–5). This is due to the layerwise displacement C0-continuitywhich allows a proper description of the stress state even between twolayers.

On the contrary, using a poor description (i.e., a single high order

element through the thickness) does not allow a correct description ofthe stress state through the thickness (see Figs. 3 and 4). In such a case,the high order continuity of the shape functions through-the-thicknessleads to the continuity of the strains through-the-thickness. Thesestrains are then multiplied by discontinuous material propertiesthrough-the-thickness to obtain the stresses. As a consequence, thestresses are discontinuous through-the-thickness (even when theyshould be continuous).

Fig. 1. Isogeometric shape functions used in the two con-sidered approaches.

Fig. 2. 3D problem used as a test case, proposed by Pagano [29].

Fig. 3. Computed stress solutions for the 3D Pagano plate problem with 3 layers at theposition =X L0.25 and =Y L0.25 . The blue solid line represents the analytical solution,black crosses represent the solution using the layerwise approach, and red circles re-present the solution using the single-element approach (without post-processing). (Forinterpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)

J.-E. Dufour et al. Composites Part B 138 (2018) 12–18

14

Fig. 6 gathers the relative computation time for these two ap-proaches with different numbers of layers. As expected, the computa-tional time of the layerwise approach is higher because the number ofdegrees of freedom is higher (1,183, 5746 and 17,407 versus 676) aswell as the number of integration points required per layer. Note thatthe computational times of the layerwise approach are, in this case,about 2.5 times higher than the single-element approach with 2 Gausspoints per layer.

The layerwise approach seems to be suitable for small stacks (i.e.,with a small amount of layers), while it is not suitable for high numberof layers as it becomes extremely time and memory consuming.

On the other hand, the single-element approach seems to be morescalable, as only one element through-the-thickness is used, but onlythe global solution in terms of displacement and in-plane stresses isaccurately captured by the simulation. The idea of the paper is to obtaina better approximation of the out-of-plane stresses by post-processingthe accurately computed displacements and in-plane stresses, withoutsignificantly increasing the overall simulation cost.

4. Reconstruction from equilibrium

Although in-plane stress components are almost correctly capturedby the single-element approach (as it can be seen in Figs. 3(b) and 4(b)for the component σ11 and 3(c) and 4(c) for the component σ12), the out-of-plane components are not. As interlaminar delamination and otherfracture processes rely mostly on such out-of-plane components, aproper through-the-thickness stress description is required. In order to

compute a more accurate stress state, we choose to use the followingpost-processing approach based on the equilibrium equations (as it hasbeen proposed in Refs. [21,22,24,26] in a finite element framework),relying on the higher regularity granted by IGA shape functions.

Fig. 4. Computed stress solutions for the 3D Pagano plate problem with 11 layers at theposition =X L0.25 and =Y L0.25 . The blue solid line represents the analytical solution,black crosses represent the solution using the layerwise approach, and red circles re-present the solution using the single-element approach (without post-processing). (Forinterpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)

Fig. 5. Computed displacement solutions for the 3D Pagano plate problem with 3 and 11layers at the position =X L0.25 and =Y L0.25 . The blue solid line represents the analy-tical solution, black crosses represent the solution using the layerwise approach, and redcircles represent the solution using the single-element approach. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version of thisarticle.)

Fig. 6. Relative computational time with respect to the number of layers. The single-element approach (red and black lines, using 4 and 2 integration points per layer re-spectively) is less time consuming than the simulation with one element per layer (blueline). In this case, there are 9 quartic elements along each in-plane direction and cubicshape functions through the thickness, corresponding to 1,183, 5746 and 17,407 degreesof freedom for the layerwise approach, and 676 for the single-element approach. (Forinterpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)

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In an equilibrium state, the stresses inside the material should sa-tisfy the equilibrium equation

∇⋅ =σ b (6)

where ∇⋅ is the divergence operator. In a 3D case, this reads in com-ponents as

+ + =+ + =+ + =

σ σ σ bσ σ σ bσ σ σ b

11,1 12,2 13,3 1

12,1 22,2 23,3 2

13,1 23,2 33,3 3 (7)

with

=∂∂

σσx

.ij kij

k,

By integrating along the thickness, given an arbitrary X3 we recoverthe out-of-plane stresses for = …i 1, ,3 as

∫= − +

− +

σ X X X σ X X ζ σ X X ζ

b X X ζ ζ σ X X X

( , , ) ( ( , , ) ( , , )

( , , ))d ( , , ),

i X

Xi i

i i

3 1 2 3 1,1 1 2 2,2 1 2

1 2 3 1 2 3

3

3

(8)

In particular σ13 and σ23 can be obtained from σ11, σ12 and σ22 bydirect integration.

∫= − + − +σ X σ ζ σ ζ b ζ ζ σ X( ) ( ( ) ( ) ( ))d ( ),X

X13 3 11,1 12,2 1 13 3

3

3

(9a)

∫= − + − +σ X σ ζ σ ζ b ζ ζ σ X( ) ( ( ) ( ) ( ))d ( ),X

X23 3 12,1 22,2 2 23 3

3

3

(9b)

∫= − + − +σ X σ ζ σ ζ b ζ ζ σ X( ) ( ( ) ( ) ( ))d ( ),X

X33 3 13,1 23,2 3 33 3

3

3

(9c)

It is important to note that, by introducing equations (9a) and (9b)into (9c), the component σ33 can then be computed as

∫ ∫

∫

= ⎡⎣+ + −

− ⎤⎦+

+ − ++

σ X σ ξ σ ξ σ ξ b ξ

b ξ ξ ζ

b ζ ζ

X X σ X σ Xσ X

( ) ( ( ) ( ) 2 ( ) ( )

( ))d d

( )d

( )( ( ) ( ))( ).

XX

Xζ

XX

33 3 11,11 22,22 12,12 1,1

2,2

3

3 3 13,1 3 23,2 3

33 3

33

3

33

(10)

The integral constants should be chosen to fulfill the boundaryconditions at the top or bottom surfaces. It can be noted that new stressboundary conditions arise from these equations. These boundary con-ditions can, in general, be computed even for more complex loadingcases than the one presented herein.

Remark 1. It should be noted that the integration could be per-formed from both surfaces and then averaged in order to divide theresulting error by two. Although it is easy to compute the boundaryconditions in this case, we chose to avoid this as it relies on theperfect knowledge of the stress boundary conditions on both sur-faces (including the loaded one) which are not always available.

Assuming that the elasticity tensor of equation (1) is constant, thederivatives of the in-plane components of the stress are computed fromdisplacements as

==

σ εσ ε

,,

ij k ijmn mn k

ij kl ijmn mn kl

, ,

, ,

(11)

where

= +

= +

ε u u

ε u u

( ),

( ).

mn k m nk n mk

mn kl m nkl n mkl

,12 , ,

,12 , , (12)

Equation (12) clearly demonstrates the necessity of a highly regulardisplacement solution in order to recover a proper stress state. Such

condition can be easily achieved using IGA with B-spline shape func-tions of order >p 2 while in a standard finite element framework,continuity of the solution must be enhanced (see Refs. [24,26]). Onecould thus retrieve a good approximation of the out-of-plane stress stateonce an accurate description of the in-plane stress state is available.

The displacement solution computed using an IGA method is aspline, which means that its post-processing and further handling of itsderivatives can be efficiently performed. Splines can be indeed easilydifferentiated and many dedicated tools are available [31].

4.1. Numerical results

Figs. 7–9 show that applying the post-processing allows to greatlyimprove the quality of the results. Indeed, we observe that the post-processed stress state is very close to the analytical one. The correctstress state is recovered on the whole plate, and a great improvement isobserved everywhere. The results were obtained using an in-plane meshcomposed of ×9 9 quartic elements with cubic shape functions throughthe thickness.

In order to validate such an approach in a wider variety of cases,computations with several sets of number of layers for the laminate or adifferent ratio between the thickness of the plate and its length can beperformed. Fig. 10 gathers the results for different cases (varying thenumber of layers) with respect to the length-thickness ratio S used inthe computations. In each layer, a standard Gaussian integration with

+p 1 points (i.e., 5, in our case) along each in-plane direction has beenused. The error is computed along the stiffest direction (i.e., 23 in thiscase) as:

=−σ σ

σerror

max( )max( )

.analytic recov

analytic (13)

Several conclusions can be drawn from Fig. 10. First, the post-pro-cessing method provides better results when the thickness ratio islarger. The method should therefore be used on slender plates ratherthan on thick stacks. Concerning the integration scheme, +p 1 pointsper layer provide basically the same accuracy as using −p 1 points perlayer (i.e., 2 points, given the cubic through-the-thickness

Fig. 7. Recovered (red solid line) σ13 compared to the analytical one (blue crosses) forseveral in plane positions. L is the total length of the plate, that in this case is =L 110 mm(being =L S t with =t 11 mm and =S 10), while the number of layers is 11. (For in-terpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)

J.-E. Dufour et al. Composites Part B 138 (2018) 12–18

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approximation). −p 2 integration points per layer (corresponding tojust 1 point in our case) lead in general to a loss of accuracy except, inour example, in the case of 34 layers (Fig. 10(c)) where some super-convergence effect is present. Based on our tests, this “super-con-vergence” seems to be a special case happening only when a largeunsymmetrical stack is involved.

Finally, the more layers in the stack, the better the results, since astack with a large number of thin layers is indeed closer to a plate with

average properties, whose solution would be perfectly captured by astandard isogeometric simulation with one high-order element throughthe thickness.

Therefore, the post-processing approach seems to be particularlysuited to handle large and thin plates with a significant number oflayers (whereas the first approach with one element per layer is far toocostly to handle these kinds of problems, due to the huge number ofelements and degrees of freedom). In such situations, this methodprovides results very close to the correct solution, at a fraction of thecost, representing a far better choice than a full 3D computation interms of accuracy-to-cost ratio.

Although the results in this paper are obtained using a 9 × 9 ele-ments in-plane mesh, Fig. 11 shows that even using a coarse elementmesh, the error remains acceptable in this case.

5. Conclusion

In this paper, we have presented a very simple approach for anaccurate simulation of laminates combining a standard 3D coarse iso-geometric analysis with a layerwise integration rule and a post-pro-cessing based on equilibrium equations. Using this approach, resultsclose to the analytical solution are obtained, while using only few de-grees of freedom through the thickness in contrast with a standard 3Dlayerwise approach.

Multiple numbers of layers (both even and odd) have been con-sidered in our numerical tests and different thicknesses and numbers ofintegration points per layer have been studied. Whatever the number oflayers, the method gives better results the thinner the layers are. Basedon our results, a through-the-thickness integration rule involving −p 1points per layer can be considered as the default choice, since it grants agood accuracy even when the plate becomes thicker.

This method is easy to implement. Indeed, the post-processing isonly based on the integration of equilibrium equations, and all the re-quired components can be easily computed from the displacement so-lution. Furthermore, a standard 3D isogeometric computation is used,with a special set of integration points, but without any modification inthe model. As the IGA code provides spline fields, its derivation andhandling are straightforward and the whole process (coarse simulationplus post-processing) is far less time consuming than a full layerwise 3D

Fig. 8. Recovered (red solid line) σ23 compared to the analytical one (blue crosses) forseveral in plane positions. L is the total length of the plate, that in this case is =L 110 mm(being =L S t with =t 11 mm and =S 10), while the number of layers is 11. (For in-terpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)

Fig. 9. Recovered (red solid line) σ33 compared to the analytical one (blue crosses) forseveral in plane positions. L is the total length of the plate, that in this case is =L 110 mm(being =L S t with =t 11 mm and =S 10), while the number of layers is 11. (For in-terpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)

Fig. 10. Difference between post-processed and analytical stress state. Different numbersof integration points per layer along the thickness (nquad) and length-to-thickness ratios S

are used.

J.-E. Dufour et al. Composites Part B 138 (2018) 12–18

17

approach.Further research topics currently under investigation involve the

extension of this approach to more complex problems involving curvedgeometries and/or large deformations. In such cases, specific care isrequired to integrate the equilibrium equations since the curvilinearform of the divergence operator leads to a non linear partial derivativeproblem to be solved. Additionally, a new methodology based on theuse of the sum factorization technique [32] is being explored. Thiswould allow to reduce drastically the assembling time both for thesingle-element and the layerwise approaches. Moreover, more efficientthrough-the-thickness integration strategies or the combination withbivariate shells will be also considered in future studies.

Acknowledgment

P. Antolin was partially supported by the European ResearchCouncil through the H2020 ERC Advanced Grant 2015 n.694515CHANGE.

G. Sangalli was partially supported by the European ResearchCouncil through the FP7 ERC Consolidator Grant n.616563 HIGEOM.

J.-E. Dufour, F. Auricchio, and A. Reali were partially supported byFondazione Cariplo - Regione Lombardia through the project “Versonuovi strumenti di simulazione super veloci ed accurati basatisull'analisi isogeometrica”, within the program RST - rafforzamento.

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Fig. 11. Convergence of the in-plane stress solution with respect to the number of in-plane elements along each direction with =S 10. The degree of the shape functions re-mains 4 in-plane and 3 through the thickness.

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