elements of voluminal hull shb with 6.15 and 20 noeu []

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Titre : Éléments de coque volumique SHB à 6, 15 et 20 noeu[...] Date : 30/09/2013 Page : 1/42Responsable : DE SOZA Thomas Clé : R3.07.08 Révision :

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Elements of voluminal hull SHB with 6.15 and 20nodes

Summary:

We present in this document 3 new elements of voluminal hull intended to supplement modeling SHB, whichcomprises already the element SHB8 having for mesh support a hexahedron with 8 nodes [R3.07.07]. These 3elements are:

• the element SHB6 who has as a mesh support a pentahedron with 6 nodes,• the element SHB15 who has as a mesh support a pentahedron with 15 nodes,• the element SHB20 who has as a mesh support a hexahedron with 20 nodes.

Just as it SHB8, these 3 elements have a called privileged direction thickness. Thus, they can be used torepresent mean structures while correctly taking into account the phenomena through the thickness (inflection,elastoplasticity), grace a digital integration to 5 points of Gauss in this privileged direction.

Like SHB8, and in order to reduce time calculation, these elements under-are integrated but, contrary to him,they do not have modes of Hourglass (modes of deformation to worthless energy) and thus do not require amechanism of stabilization. Nevertheless, to avoid blockings (in particular in transverse shearing), it SHB6 isproject following the method of the supposed deformations (assumed strain).The quadratic elements neither are stabilized, nor projected.

In addition to their cost of relatively weak calculation and their good performances in elastoplasticity, theseelements have another advantage. Since they are based on a three-dimensional formulation and that they haveonly degrees of freedom of translation, it is easy to couple them with voluminal elements 3D, which is veryuseful in systems where voluminal hulls and elements must cohabit.

Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in partand is provided as a convenience.Copyright 2021 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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Contents1 Introduction........................................................................................................................................... 5

2 Element SHB6...................................................................................................................................... 6

2.1 Kinematics of the element..............................................................................................................6

2.2 Discretization................................................................................................................................. 6

2.2.1 Discretization of the field of displacement.............................................................................6

2.2.2 Operator discretized gradient................................................................................................7

2.3 Matrix of rigidity and stabilization...................................................................................................9

2.3.1 Matrix of rigidity..................................................................................................................... 9

2.3.2 Analysis of the modes “hourglass” for element SHB6.........................................................10

2.3.3 Projection by “local Assumed strain method”......................................................................11

2.4 Geometrical matrix of rigidity Ksigma...........................................................................................12

2.5 Following forces and matrix of pressure Kp.................................................................................14

3 Elements SHB15 and SHB20.............................................................................................................16

3.1 Kinematics and interpolation of elements SHB15 and SHB20.....................................................16

3.1.1 Element SHB15.................................................................................................................. 16

3.1.2 Element SHB20.................................................................................................................. 17

3.2 Operator discretized gradient.......................................................................................................18

3.2.1 Element SHB15.................................................................................................................. 18

3.2.2 Element SHB20.................................................................................................................. 24

3.3 Variational formulation used for elements SHB15 and SHB20.....................................................30

3.4 Geometrical matrix of rigidity Ksigma...........................................................................................33

3.5 Following forces and matrix of pressure Kp.................................................................................35

4 Strategy for non-linear calculations....................................................................................................37

4.1 Geometrical non-linearities..........................................................................................................37

4.2 Non-linearities materials...............................................................................................................38

5 Establishment of elements SHB in Code_Aster..................................................................................40

5.1 Description................................................................................................................................... 40

5.2 Use............................................................................................................................................... 40

5.2.1 Grid..................................................................................................................................... 40

5.2.2 Modeling............................................................................................................................. 40

5.2.3 Material............................................................................................................................... 40

5.2.4 Boundary conditions and loading........................................................................................40

5.2.5 Calculation in linear elasticity..............................................................................................40

5.2.6 Calculation in linear buckling..............................................................................................40

5.2.7 Calculation in geometrical nonlinear “elasticity”..................................................................40

5.2.8 Calculation nonlinear plastic...............................................................................................41

5.3 Establishment.............................................................................................................................. 41

5.4 Validation..................................................................................................................................... 41Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in partand is provided as a convenience.Copyright 2021 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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6 Bibliography........................................................................................................................................ 42




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1 Introduction

Element of a solid type - voluminal hull of hexahedral geometry at five points of Gauss was alreadyestablished in ASTER. Good performances of this element, named SHB8, were put in obviousness byAbed-Meraim and Combescure [bib1], [bib2] like by Legay in [bib3]. This element represents a thickhull obtained starting from a purely three-dimensional formulation. It has eight nodes and five points ofintegration distributed according to the direction thickness. The three-dimensional law of behavior wasalso amended to approach the behavior of the hulls and to avoid certain lockings (shearing,membrane). To eliminate the modes with worthless energy due to under-integration, an effectivetechnique of stabilization was used while following the approach of Belytschko and Bindeman [bib4]. Inthe same way, the operator discretized gradient was modified for the elimination of various blockings.Thus, the version obtained of this element has the following advantages:

• capacity to model mean three-dimensional structures with few elements of grid thanks to thetolerated important twinge (significant time-saver of calculations),

• simplified grid of complex geometries where solid hulls and elements must cohabit (reinforcementsor supports for example) without having the classical problems of connections of made grids ofvarious types of elements.

This hexahedral element was introduced into Code_Aster in version 7 (see [R3,07,07]). However, thehexahedral element SHB8 does not allow to net geometries of complex forms unspecified. Thedevelopment of a similar element but of prismatic geometry was thus necessary. One describes in thebeginning of this document this prismatic element (element SHB6).

The research tasks of Caironi and Abed-Meraim [bib5] proved that the element SHB6 did not presentmodes of hourglass, and after having established it, they as showed as this one presented a severedigital blocking, in particular in the requests in transverse shearing of the element. The element SHB6established in Aster these digital blockings by using the method “assumed strain aims at eliminating”.The principle of this method consists in projecting the operator discretized gradient B on a suitablesubspace in order to avoid the various problems involved in blocking. Several projections were testedbefore finding that which eliminates the maximum of lockings.

The element SHB6 fact the object of the §2.

The §3 presents an extension of this family of finite elements of standard solid-hull: two finite elementsof prismatic and hexahedric geometry but of quadratic formulation named SHB15 and SHB20. They arerespectively elements with 15 and 20 nodes. They under-are also integrated by 15 and 20 points ofGauss and have a direction privileged according to the thickness of the element. These elements nothaving blockings are not projected.

In addition, the element SHB8 initial had been coupled with the only laws of elastic and elastoplasticbehaviour with isotropic work hardening of type Von-Put. The field of application of the element SHB8as well as the other finite elements solid-hull SHB6, SHB15 and SHB20 was extended to the otherlaws of behavior of Code_Aster. The §4 presents the theoretical principle of this coupling.

Finally the §5 treats establishment of these elements in Code_Aster.




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2 Element SHB6

2.1 Kinematics of the element

Element SHB6 is a pentahedron with 6 nodes. The five points of integration are selected along thedirection in the reference mark of the local coordinates of the element of reference: , , (orx1, x2, x3 for certain expressions). The shape of the element of reference as well as the points of

integration are represented on [Figure 1]. x1 or

5

1 2 3 4

1

2 3

4

5 6

O x3 ou

x2 or

Figure 1: Geometry of the element of reference and points of integration

This element is isoparametric and has the same linear interpolation and same kinematics as thepentahedral elements with 6 standard nodes.

2.2 Discretization

2.2.1 Discretization of the field of displacement

Space coordinates x i element are connected to the nodal coordinates x iI by means of the

isoparametric functions of forms N I by the formulas:

x i=x iI N I x1, x2, x3=∑i=1

6

x iI N I x1, x2, x3

In the continuation, and except contrary mention, one will adopt the convention of summation for therepeated indices. Indices in small letters i vary from one to three and represent the directions of thespace coordinates. Those in capital letters I vary from one to six and correspond to the nodes of theelement.




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The same functions of forms are used to define the field of displacement of the element iu according

to nodal displacements u iI :

ui=U iI N I x1, x2, x3=∑i=1

6

U iI N I x1, x2, x3

To continue calculations, linear isoparametric functions of form are given N i , ,=N i x1, x2, x3associated with the prismatic element with six nodes:

1323

132

121

112

1

12

1

12

1

xxxN

xxN

xxN

1326

135

124

112

1

12

1

12

1

xxxN

xxN

xxN

]1,0[];1,0[];1,1[ 2321

xxxx

The origin of the reference mark is confused with the right corner of the triangle of the median planeof the element.

2.2.2 Operator discretized gradient

The gradient u i , j field of displacement is a function of displacements iIU nodes I in the direction i

i j iI I ju N, , U

The linear tensor of deformation is given by the symmetrical part of the gradient of displacement:

1( )

2ij i j j iu u

One now will build vectors allowing to express the matrix B connecting the deformations todisplacements in a particular form.

In a way similar to Belytschko-Bindeman [bib6], the three vectors are introduced ib , derived from the

functions of form at the origin of the coordinates:

biT=N , i 0=

∂N∂ xi ]x 1= x 2=x3=0

i=1,2,3

These 3 vectors are constant and are given by the expression:




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biT= j i1 ji2 j i3 ⋅[

0 0 −12

0 0 12

12

0 −12

12

0−12

012

−12

012

−12]

where coefficients klj are the coefficients of the matrix jacobienne evaluated in the beginning.

Also let us introduce the following vectors:

Ts = ( 1 1 1 1 1 1 )T1h = ( -1 0 0 1 0 0 )T2h = ( 0 -1 0 0 1 0 )

),,,,,( 654321 iiiiiiTi xxxxxxX

Three vectors TiX the nodal coordinates of the six nodes represent. Two vectors

Th the functions

represent respectively 1h and 2h for each of the six nodes, which are defined by:

211 xxh

312 xxh

Let us introduce finally the two following vectors:

3

1

.2

1

jjj

T bXhh 2,1

One can check by algebraic considerations that the following conditions of orthogonality are satisfied:

0 hbTi

0sbTi

ijjTi Xb 3,2,1, ji 2,1,

2hh T (1)

0 jT X

hT

where ij is the symbol of Kronecker.

These vectors will make it possible to express the matrix B connecting the deformations todisplacements in a particular form used thereafter.


0sh T



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The gradient of the field of displacement can be now written after calculations in the form (without anyapproximation [bib6]):

ui , j=b jTh , jT ⋅U i

The symmetrical operator gradient (noted s ) discretized connecting the tensor of deformation to the

vector of nodal displacements

∇ s u=B⋅u takes the matric shape then:

0 0

0 0

0 0

0

0

0

T Tx x

T Ty y

T Tz z

T T T Ty y x xT T T Tz z x x

T T T Tz z y y

h

h

h

h h

h h

h h

b

b

bB

b b

b b

b b

(2)

2.3 Matrix of rigidity and stabilization2.3.1 Matrix of rigidity

The matrix of rigidity of the element is given by:

K e=∫V e

BT⋅C⋅B dV

Five points of integration considered iP are on the same vertical line. Their coordinates and their

weights of integration are the following:

2x

3x

1x

1P 1/3 1/3 -0.906179845938664 0.236926885056189

2P 1/3 1/3 -0.538469310105683 0.478628670499366

3P 1/3 1/3 0 0.568888888888889

4P 1/3 1/3 0.538469310105683 0.478628670499366

5P 1/3 1/3 0.906179845938664 0.236926885056189

Thus, the expression of rigidity eK is:

)(.).()()(5

1jj

Tj

jje PBCPBPJPK

(3)

where )( jPJ is Jacobien, calculated at the point of Gauss j , transformation enters the element of

reference and the current element. The elastic matrix of behavior C chosen has the following form:




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2 0 0 0 0

2 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

E

C

where E is the Young modulus, the Poisson's ratio,

12

E the modulus of rigidity and

21

E

the coefficient of modified Lamé. This law is specific to elements SHB. It resembles that

which one would have in the case of the assumption of the plane constraints, put except for the term(3.3).

Even if this choice involves an artificial anisotropic behavior, it makes it possible to satisfy all the testswithout introducing blocking.

2.3.2 Analysis of the modes “hourglass” for element SHB6

The modes of “hourglass” are kinematics modes which are due to under-integration and areassociated with a worthless energy whereas they induce a nonworthless deformation. This anomaly isexplained by the difference, that induced under-integration, between the core of the continuousoperator of rigidity discretized and that. Let us start initially by noticing that the operator discretizedgradient under-integrated associated with the five points of integration defined above takes the shapeof the equation (2) with 2,1 .

Now let us analyze the core of the matrix of rigidity obtained by under-integration. According to (3), thatreturns under investigation from the row of the matrix B insofar as the matrix of behavior C is notsingular. In other words, it is enough to search the modes of displacement d with worthlessdeformation, i.e. checking:

∇ s u=B⋅d=0 (4)

We will seek from now on which are the modes of deformations which give a worthless deformation

energy. The deformation energy is written w =12∫V

⋅C⋅ dV and like =B .d . we thus have:

w =12∫VdT BT⋅C⋅B d dV=d T [ 1

2∫VBT⋅C⋅B dV ] d

and if we consider the following approximation: B is calculated at the points of integration of Gauss,we obtain:

w =12d T K e d

Thus to search the modes of deformations to worthless energy is to search the core of K e

K e⋅X=0⇔ BG j⋅X=0

Thus to search the modes of hourglass is to search the vectors X such as:




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BG j⋅X=0 ∀G j (5)

It is natural to find in the core of rigidity K e modes associated with the rigid movements of body. For athree-dimensional element such as the prism with 6 nodes, these movements rigidifying are composedof three translations and three rotations. Thus the core of the continuous operator of rigidity is ofdimension six and is reduced to the only following modes:

et

y z 0S 0 0

0 S 0 -x 0 z

0 0 S 0 -x -y

(6)

One easily checks that each of the six vectors columns above satisfy the equation (5) and thus belongsto the core of K e . It is enough, to see it, to use the expression (2) of B and conditions oforthogonality (1). The first three vectors columns correspond to the translations according to the axesOx , Oy and Oz respectively. The three other vectors are relating to rotations around the axes Oz ,Oy and Ox .

We search from now on, in addition to the preceding rigid modes, of the modes which also cancel theoperator discretized gradient given in (2). Let us take a base of eighteen following vectors:

One can show easily that the vectors above are linearly independent within the space of dimensioneighteen. Elementary calculations using the conditions of orthogonality (1) show that the last twelvevectors columns do not check the equation (5).

That wants to say that there are not other modes only the rigid modes which cancel the operatordiscretized gradient given in (2). In other words, the element SHB6 do not present hourglass mode.

2.3.3 Projection by “local Assumed strain method”

The first stage is to place itself in the local reference mark of the element defined by the reference

mark ),,( 3

^

2

^

1

^

xxx described in Figure 1. The deformations from now on will thus be calculated in this

reference mark. The operator discretized gradient B will be project on under suitable space in orderto avoiding the various problems of blocking. This method is variationnellement coherent with theprinciple of Hu-Washizu if the interpolation of the constraint is judiciously selected (Simo and Hughes[15]). However, it is very difficult to select in a general and systematic way the good field of applieddeformation. The fields of applied deformation should present neither voluminal blocking nor blockingin shearing.

We present here an easy choice and acceptable. The operator B is first of all separate in two parts

1B and 2

B . The matrix 1B contains the gradients in the average plan of the hull and the perpendicular

deformation, 2B contains the gradients associated with the shearing strains transverse.




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The blockings noted in the element come from transverse shearings. One will seek a diagram ofintegration which allows of under - to integrate this part of energy. With this intention one seeks tocontrol each component entering the energy of transverse shearing. Being given the shape of the

matrix B we thus have 12 nonworthless terms which intervene in the deformation. They will be

controlled by the introduction of the parameter c in the matrices 2B . The matrix 2

B becomes

then 2

B :

The matrix of rigidity is written now:

K e=∫VBT⋅C⋅BdV=∫V

B1T⋅C⋅B1dV∫V B2

T⋅C⋅ B1dV ∫VB1T⋅C⋅ B2dV∫V B2

T⋅C⋅B2dV=

=K e1K e2K e3K e4

Matrices K e1,K e2,K e3,K e4 are integrated with the five points of Gauss defined previously. Additive

decomposition given higher, 21BBB , for the operator discretized gradient, makes that the cross

terms 2eK and 3eK cancel themselves. Following many the test digital, it was selected to

characterize the matrix 2

B by the coefficient: 45,0c , which plays here the part of a factor of

reduction of shearing.

This choice gives to the element a good behavior in the cases of reference. It is clear that this strategy,as that installation for the cubic elements voluminal hulls are adapted only to the quasi isotropicbehavior of selected material.

2.4 Geometrical matrix of rigidity Ksigma




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The matrix K aims to solve the problems of buckling. We point out here that the modes of bucklingare the clean vectors of the problem to the eigenvalues generalized according to:

K K ⋅u=0⇔ K⋅u= K⋅u

with , and is the multiplying coefficient of the loading.

By introducing the quadratic deformation Qe such as:

e ijQu , u=∑

k=1

3

uk ,i . uk , j

One can define this matrix of geometrical rigidity by:

uT⋅K⋅ u=∫0

: eQ u , ud=∫ 0

:∇uT∇ u d

In order to express this matrix in discretized space, let us introduce the discretized operators quadratic

gradient QB (in matric notation) such as:

Various terms Q

ijB are given by the following equations:




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with the vectors iB ( I = 1,2,3) defined as:

Note: We must multiply the matrices 13

QB and 23

QB by the coefficient 2 20,45 0,2025c because

element SHB6 is project by the technique “ Local Assumed strain method “ to see section 2.3.3 .

With these notations, the contribution to the geometrical matrix of rigidity, k , at the point of Gauss

j is given by:

11 22 3311 22 33

12 13 2312 13 23

Q Q Qj j j j j j j

Q Q Qj j j j j j

k B B B

B B B

By integration on the points of Gauss of the element, the geometrical matrix of rigidity is obtained bythe formula:

K =∑ j=1

5 j J jk j

2.5 Following forces and matrix of pressure Kp

The following compressive forces are present in the tangent matrix via the matrix PK , because the

following external forces depend on displacement [R3.03.04]. The following compressive forces arewritten:

∫∂pnT⋅udS=∫∂ 0

p det [F u]n0T F u -TdS 0= p F0− p K p⋅u

F u=1∇ u by using the notations:

•0 1 2 3( )T n n n n , normal on the surface external of the element in the configuration of

reference;

• ib%, vector of size 3, derived from the functions of form to the 3 nodes of the face of the element

charged in pressure;

• 0S surface of the face charged in pressure. For the element SHB6, this surface 0S is worth 1

2.

The preceding formulation leads to a not-symmetrical matrix. It is known that one can nevertheless usea symmetrical formulation if the external forces due to the pressure derive from a potential. It is thecase if the compressive forces do not work on the border of the modelled field. It is thus consideredthat the symmetrical part of the matrix is enough. The symmetrized matrix takes the following shape:




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K p=S0[0 b2

T n1−b1T n2

b3T n1−

b1T n3

0 b2T n1−

b1T n2

b3T n1−

b1T n3

0 b2T n1−

b1T n2

b3T n1−

b1T n3

b1T n2−

b2T n1 0 b3

T n2−b2T n3

b1T n2−

b2T n1 0 b3

T n2−b2T n3

b1T n2−

b2T n1 0 b3

T n2−b2T n3

b1T n3−

b3T n1

b2T n3−

b3T n2 0

b1T n3−

b3T n1

b2T n3−

b3T n2 0

b1T n3−

b3T n1

b2T n3−

b3T n2 0

]

It is a matrix (9, 9), that it is necessary to multiply by displacements of the 3 nodes of the face to which one applies a pressure.




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3 Elements SHB15 and SHB20

In this paragraph, one presents modelings of the finite elements quadratic voluminal hulls SHB15 andSHB20.

The element SHB15 is a purely three-dimensional prism with fifteen nodes with three degrees offreedom in displacement to each node, and it also has a called privileged direction “thickness” which isnormal with the average plan of the prism. Reduced digital integration is used (3 points of Gauss in theplan). Integration through the thickness is based on 5 points of Gauss.

The element SHB20 is a purely three-dimensional hexahedron with twenty nodes with three degrees offreedom in displacement to each node, and it has also a called privileged direction “thickness” which isnormal with the average plan of the hexahedron. Reduced digital integration is used (4 points of Gaussin the plan). Integration through the thickness is based on 5 points of Gauss.

Contrary to the linear elements these finite elements have neither stabilization nor projection.

3.1 Kinematics and interpolation of elements SHB15 and SHB20

3.1.1 Element SHB15

The element SHB15 is formulated in the local axes of the average plan. F igure 3.1.1-a represent thegeometry of an element of reference SHB15 and its points of integration.

The reference mark of the local coordinates of the element of reference is defined by: , , ou x1, x2, x3

x1=

5

1

2 3

4

1

2

3

4

5 6

O

7

8

9

1 0

1 1 1 2

1 0

6 7

8 9

1 5

1 1

1 2 1 3

1 4

1 3

1 4

1 5

x3=

x2= F igure 3.1.1-a . Geometry of the element of reference SHB15 and its points of integration




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Coordonnées des noeuds:

1 1,1,0 ; 2 1,0,1 ; 3 1,0,0 ; 4 1,1,0 ; 5 1,0,1 ; 6 1,0, 0 ;

1 1 1 17 1, , ; 8 1,0, ; 9 1, ,0 ;

2 2 2 2

1 1 1 110 0,1,0 ; 11 0,0,1 ; 12 0,0,0 ; 13 1, , ; 14 1,0, ; 15 1, ,0 .

2 2 2 2

Element SHB15 is an isoparametric quadratic element. Space coordinates ix are connected to the

nodal coordinates iIx by means of the functions of form IN by the formulas:

x i=x iI N I x1, x 2, x3=∑i=1

15

x iI N I x1, x2, x3

The same functions of form are used to define the field of displacement of the element iu in terms of

nodal displacements U iI : :

ui=u iI N I x1, x2, x3=∑i=1

15

U iI N I x1, x2, x3 (6)

3.1.2 Element SHB20

Element SHB20 is formulated in the local axes of the average plan. Figure 3.1.2-a represent thegeometry of an element of reference SHB20 and its points of integration.The reference mark of the local coordinates of the element of reference is defined by: , , ou x1, x2, x3

x2=

5

1

2

3 4

1

2 3

4

5

6 7

8

1 0

9 1 1

1 4 1 5

1 6

2 0

1 9

1 8

1 7

1 3

1 2

6 7

8

9

1 0

1 1

1 2 1 3

1 4

1 5

1 6 1 7

1 8

1 9 2 0

x3=

x1=

Figure 3.1.2-a. Geometry of the element of reference SHB20 and its points of integration




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Coordonnées des noeuds:

1 1, 1, 1 2 1, 1, 1 3 1,1, 1 4 1,1, 1

5 1, 1,1 6 1, 1,1 7 1,1,1 8 1,1,1

9 0, 1, 1 10 1,0, 1 11 0,1, 1 12 1,0, 1

13 1, 1,0 14 1, 1,0 15 1,1,0 16 1,1,0

17 0, 1,1 18 1,0,1 19 0,1,1 20 1,0,1

Element SHB20 is also an isoparametric quadratic element. Space coordinates ix are connected to

the nodal coordinates iIx by means of the functions of form IN by the formulas:

x i=x iI N I x1, x 2, x3=∑i=1

20

x iI N I x1, x2, x3

The same functions of form are used to define the field of displacement of the element iu in terms of

nodal displacements U iI :

ui=u iI N I x1, x2, x3=∑i=1

20

U iI N I x1, x2, x3 (6)

3.2 Operator discretized gradient

3.2.1 Element SHB15

The interpolation of the field of displacement of the element (6) will allow us to define the rate ofdeformation and to write the relations connecting the deformations to nodal displacements. One starts

initially by writing the gradient ,i ju field of displacement:

ui , j=U iI N I , j (7)

The tensor of deformation ij is given then by the symmetrical part of the gradient of displacement:

1( )

2ij i j j iu u (8)

To continue calculations, quadratic isoparametric functions of form are given N i x1, x2, x3 , associatedwith the prismatic element with fifteen nodes:

11323

13132

12121

112

1

)22(12

1

)22(12

1

xxxxN

xxxxN

xxxxN

11326

13135

12124

112

1

)22(12

1

)22(12

1

xxxxN

xxxxN

xxxxN

(9)




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1(27N

1x )

2x

3x 1(28N

1x )

321 xx

3x 1(29N

1x )

2x

321 xx

1(10N

1x )^

1)1( x

2x 1(11N

1x )^

1)1( x

3x 1(12N

1x )^

1)1( x

321 xx

1(213N

1x )

2x

3x 1(214N

1x )

321 xx

3x 1(215N

1x )

2x

321 xx

]1,0[];1,0[];1,1[ 2321

xxxx

While combining the preceding equations one manages to develop the field of displacement as being

the sum of a constant term, linear terms in ix , and of terms utilizing functions h

To simplify the writings, one will note =x1 , =x2 , =x3

0 1 1 2 2 3 3 1 1 2 2 3 3

4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11

2 21 2 3 4 5 6

2 2 2 2 27 8 9 10 11

u h h h

h h h h h h h h

1,2,3

h ,h , h ,h , h ,h ,

h ,h ,h , h , h

i i i i i i i i

i i i i i i i i

a a x a x a x c c c

c c c c c c c c

i

(10)

By evaluating the equation (6) at the nodes of the element, one arrives at the three systems of fifteenequations following:

1 2 3 1 2 30 1 2 3 1 2 3

4 5 6 7 8 9 10 114 5 6 7 8 9 10 11

1,2,3

i i i i i i i i

i i i i i i i i

a a a a c c c

c c c c c c c c

i

d S x x x h h h

h h h h h h h h (11)

Thus vectors id and ix represent, respectively, displacements and the nodal coordinates and are

given by:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

, , , , , , , , , , , , , ,

, , , , , , , , , , , , , ,

Ti i i i i i i i i i i i i i i i

Ti i i i i i i i i i i i i i i i

u u u u u u u u u u u u u u u

x x x x x x x x x x x x x x x

d

x(12)

Vectors S and h 1,2,3,...,11 are given as for them by:




c2b0aa7ea6ee

1

2

3

4

5

6

7

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 10 1 0 0 0 0 0 0 1 0 0

2 2 2 2

1 1 1 10 0 0 1 0 0 0 0 0 0 1

2 2 2 2

1 10 0 0 0 0 0 0 0 0 0 0 0 0

4 4

1 10 0 0 0 0 0 0 0 0 0 0 0 0

4 4

1 1 1 10 1 0 0 0 1 0 0 1 0 0

4 4 4 4

1 1 1 10 0 0 1 0 0 1 0 0 0 1

4 4 4 4

1

T

T

T

T

T

T

T

T

S

h

h

h

h

h

h

h

8

9

10

11

1 1 1 1 1 0 0 0 1 1 1 1 1 1

1 1 1 10 1 0 0 0 0 0 0 1 0 0

4 4 4 4

1 1 1 10 0 0 1 0 0 0 0 0 0 1

4 4 4 4

1 1 1 10 1 0 0 0 0 0 0 1 0 0

2 2 2 2

1 1 1 10 0 0 1 0 0 0 0 0 0 1

2 2 2 2

T

T

T

T

h

h

h

h

(13)

To arrive at an advantageous writing of the operator discretized gradient B , one will introduce the three

vectors ib defined by:

)0()0(,

i

T

iTT

i x

NNb

I = 1,2,3 (14)

If we place ourselves in )0,0,0(),,( 3

^

2

^

1

^

xxx then we obtain:

csteNb iTT

i )0(,

where TN represent: 1 2 3 15N N N . . N .

))0(x

N),.....,0(

x

N()0(

ii

,

151

iTT

i Nb




c2b0aa7ea6ee

1 2 3j j j j 00

11

0

0

N N N N N N N. . . . . .

1,2,...,15 1,2,3

I I I I I I Ij j jj j j

x x x x

avec I et j

x y zj

x y z

x y z

F1 12 13

21 22 23

31 32 33

j j

j j j

j j j

We have:

Therefore, in )0,0,0(),,( 3

^

2

^

1

^

xxx , we have:




c2b0aa7ea6ee

1 2 3

1 2 1 0 0 0 1 1 0 1 2 1 0 0 0

. 1 0 0 0 1 2 1 0 1 1 0 0 0 1 2

1 10 0 0 0 0 0 0 0 0 0 0 0 0

2 2

Ti i i ij j j

b

Moreover, one can check by algebraic considerations that the following conditions of orthogonality are

satisfied:

1 2 3 4 5 6

7 8 9 10 11

. 0; . 0; .

1. 0; . 0; . ; . 0; . 4; . 4;

2

. 12; . 0 . 0 . 4 . 4

1 1 5 13 0 0 0 0 0 0

2 4 2 41 1 1 5

3 0 0 0 0 0 02 4 4 2

1 1 1 1 1 10 0 0 0 0

8 8 8 2 4 41 1 1 1 1

0 0 0 0 0 04 4 8 8 8

1 13 10 0 0 3 0 0

8 4 8

.

T T Tji i i ij

T T T T T T

T T T T T

Tm n

b h b S b x

h S h S h S h S h S h S

h S h S h S h S h S

h h

5 1

2 41 1 13 1 5

0 0 0 3 0 08 8 4 4 21

0 0 0 3 3 12 0 0 4 42

5 1 1 9 10 0 0 0 0 0

2 4 8 4 81 5 1 1 9

0 0 0 0 0 04 2 8 8 4

1 5 1 10 0 0 4 0 0 3

4 2 4 21 1 5 1

0 0 0 4 0 0 34 4 2 2

1, 2,...,11 , 1,2,3i j

désigne le symbol de Kro

ker ; , 1,2,...,11nec m n

(15)

This stage, one can determine the constant unknown factors which intervene in the writing (10) of the field

of displacement by multiplying scalairement the equation (11) byTjb , TS and

Th respectively, and by using the

relations of orthogonality (15).




c2b0aa7ea6ee

One obtains after calculations: .Tj ijia b d .T iic γ d

with

1 1 2 21 2

3 3 4 43 4

5 5 6 65 6

. .

1 1. .

30 30

4 4 4 4. . .

15 15 15 15

T T T T T T Tj jj j

T T T T T T T Tj jj j

T T T T T T T T Tj jj

n n

n n

n n

γ h h x b h h x b

h S h S x b h h x b

h S h S x b h S h S x

7 7 8 8 9 97 8 9

10 10 11 1110 11

4 4. . .

5 5

4 4 4 4. .

15 15 15 15

Tj

T T T T T T T T T T Tj j jj j j

T T T T T T T T Tj jj

n n n

n n

b

h S h S x b h h x b h h x b

h S h S x b h S h S x Tj

b

170 0 8 0 0 0 9 0 0 0

217

0 0 8 0 0 0 0 9 0 02

256 36 36 58 580 0 0 2 0 0

17 17 17 17 178 8 0 24 0 0 0 8 8 0 0

36 316 146 324 1710 0 0 1 0 0

17 187 187 187 18736 146 316 171 324

0 0 0 1 0 017 187 187 187 187

3 3 30 0 2 0 1 1 0 0

2 2 29 0 0 8 0 0 0 10 0 0 0

0 9 0 8 0 0 0 0 10 0 0

580 0

n , 1, 2,...,11

324 171 3 505 5850 0 0

17 187 187 2 187 37458 171 324 3 585 505

0 0 0 0 017 187 187 2 374 187

The field of displacement is put finally in the following form:




c2b0aa7ea6ee

(16)

By deriving the formula above compared to jx , the gradient of displacement is obtained:

u i , j= b jt∑=1

11

h , jT ⋅d i (17)

3.2.2 Element SHB20

The interpolation of the field of displacement of the element will enable us to define the rate of deformation andto write the relations connecting the deformations to nodal displacements. One starts initially by writing the

gradient ,i ju field of displacement:

, ,i j iI I ju u N (18)

The tensor of deformation ijε is given then by the symmetrical part of the gradient of displacement:

1( )

2ij i j j iu u (19)

To continue calculations, quadratic isoparametric functions of form are given N i x 1, x2, x3 associated withthe hexahedral element with twenty nodes:




c2b0aa7ea6ee

^

3

^

1

2^

210

^

3

^

2

2^

19

^

3

^

2

^

1

^

3

^

2

^

18

^

3

^

2

^

1

^

3

^

2

^

17

^

3

^

2

^

1

^

3

^

2

^

16

^

3

^

2

^

1

^

3

^

2

^

15

^

3

^

2

^

1

^

3

^

2

^

14

^

3

^

2

^

1

^

3

^

2

^

13

^

3

^

2

^

1

^

3

^

2

^

12

^

3

^

2

^

1

^

3

^

2

^

11

1114

1

1114

1

21118

1

21118

1

21118

1

21118

1

21118

1

21118

1

21118

1

21118

1

xxxN

xxxN

xxxxxxN

xxxxxxN

xxxxxxN

xxxxxxN

xxxxxxN

xxxxxxN

xxxxxxN

xxxxxxN

(20)

^

3

^

1

2^

220

^

3

^

2

2^

119

^

3

^

1

2^

218

^

3

^

2

2^

117

^

2

^

1

2^

316

^

2

^

1

2^

315

^

2

^

1

2^

314

^

2

^

1

2^

313

^

3

^

1

2^

212

^

3

^

2

2^

111

1114

1111

4

1

1114

1111

4

1

1114

1111

4

1

1114

1111

4

1

1114

1111

4

1

xxxNxxxN

xxxNxxxN

xxxNxxxN

xxxNxxxN

xxxNxxxN

]1,1[];1,1[];1,1[ 321

xxx

While combining the preceding equations one manages to develop the field of displacement as being the

sum of a constant term, linear terms in ix , and of terms utilizing functions h




c2b0aa7ea6ee

0 1 1 2 2 3 3 1 1 2 2 3 3 4 4 5 5 6 6 7 7

8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16

2 2 2 2 21 2 3 4 5 6 7 8 9

1

u h h h h h h h

h h h h h h h h h

1,2,3

h ,h ,h , h ,h , h , h ,h ,h ,

h

i i i i i i i i i i i i

i i i i i i i i i

a a x a x a x c c c c c c c

c c c c c c c c c

i

2 2 2 2 2 2 20 11 12 13 14 15 16, h , h ,h ,h ,h ,h

(21)

To simplify the writings, one will note =x1 , = x2 , =x3

By evaluating the equation 16 at the nodes of the element, one arrives at the three systems of twentyequations following:

1 2 3 1 2 3 160 1 2 3 1 2 3 16...

1, 2,3i i i i i i i i ia a a a c c c c

i

d S x x x h h h h (22)

Thus vectors id and ix represent, respectively, displacements and the nodal coordinates and are given by:

1 2 3 20

1 2 3 20

, , ,...,

, , ,...,

Ti i i i i

Ti i i i i

u u u u

x x x x

d

x(23)

Vectors S and h 1,2,3,...,16 are given as for them by:

1

2

3

4

5

6

7

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1

1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 0 1 0

1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0

1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1

1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0

1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1

1 1 1 1 1 1 1

T

T

T

T

T

T

T

T

S

h

h

h

h

h

h

h

8

9

10

11

12

13

14

1 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0

1 1 1 1 1 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1

1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0

1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 0 1 0

1 1 1 1 1 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1

1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 0 1 0

T

T

T

T

T

T

h

h

h

h

h

h

h

15

16

1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

T

T

T

h

h

(24)




c2b0aa7ea6ee

To arrive at an advantageous writing of the operator discretized gradient B , one will introduce the

three vectors ib defined by:

)0()0(,

i

T

iTT

i x

NNb

I = 1,2,3 (25)

If we place ourselves in )0,0,0(),,( 3

^

2

^

1

^

xxx then we obtain:

csteNb iTT

i )0(,

where TN represent: N1 N2 N3 . . .N20

))0(x

N),.....,0(

x

N()0(

i

20

i

,

1

iTT

i Nb

1 2 3j j j j 00

11

0

0

N N N N N N N. . . . . .

1, 2,..., 20 1, 2,3

I I I I I I Ij j jj j j

x x x x

avec I et j

x y zj

x y z

x y z

F1 12 13

21 22 23

31 32 33

j j

j j j

j j j

After calculations one finds:

1 2 3

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0

8 8 8 8 8 8 8 8 4 4 4 4 4 4 4 41 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

. 0 0 0 08 8 8 8 8 8 8 8 4 4 4 4 4 4 4 41 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 08 8 8 8 8 8 8 8 4 4 4 4 4 4 4 4

Ti i i ij j j

b

Moreover, one can check by algebraic considerations that the following conditions of orthogonality are satisfied:




c2b0aa7ea6ee

0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0. 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00

Tm n

12 812 8

12 816 12 1212 16 1212 12 16

812 8h h 12 8

12 88 12

0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0

, 1, 2,...,16m n

8 128 12

8 88 8

8 8

This stage, one can determine the constant unknown factors which intervene in the writing (21) field of

displacement by multiplying scalairement the equation (22) byTjb , TS and

Th respectively, and by using the

relations of orthogonality (26). One obtains:

.Tj ijia b d .T iic γ d

with:

1 1 2 2 3 31 2 3

4 4 5 54 5

6 66

. . .

4 4 4 4. .

5 5 5 5

4 4.

5 5

T T T T T T T T T Tj j jj j j

T T T T T T T T T Tj jj j

T T T Tj

n n n

n n

n

γ h h x b h h x b h h x b

h S h S x b h S h S x b

h S h S x b

7 77

8 8 9 9 10 108 9 10

11 11 12 12 13 1311 12 13

14 14 15 15 16 1614 15 16

.

. . .

. . .

. . .

T T T Tjj j

T T T T T T T T Tj j jj j j



n

n n n

n n n

n n n

h h x b

h h x b h h x b h h x b






c2b0aa7ea6ee

1 10 0 0 0 0 0 0 0 0 0 0 0 0 0

4 41 1

0 0 0 0 0 0 0 0 0 0 0 0 0 04 4

1 10 0 0 0 0 0 0 0 0 0 0 0 0 0

4 43 1 1

0 0 0 0 0 0 0 0 0 0 0 0 08 8 81 3 1

0 0 0 0 0 0 0 0 0 0 0 0 08 8 81 1 3

0 0 0 0 0 0 0 0 0 0 0 0 08 8 8

10 0 0 0 0 0 0 0 0 0 0 0 0 0 0

83 1

0 0 0 0 0 0 0 0 0 0 0 0 0 020 10

3 10 0 0 0 0 0 0 0 0 0 0 0 0 0

20 103 1

0 0 0 0 0 0 0 0 0 0 0 020 10

n ,

0 0

1 30 0 0 0 0 0 0 0 0 0 0 0 0 0

10 201 3

0 0 0 0 0 0 0 0 0 0 0 0 0 010 20

1 30 0 0 0 0 0 0 0 0 0 0 0 0 0

10 201 1

0 0 0 0 0 0 0 0 0 0 0 0 0 04 8

1 10 0 0 0 0 0 0 0 0 0 0 0 0 0

4 81 1

0 0 0 0 0 0 0 0 0 0 0 0 0 04 8

1,2,...,16

For 2 elements SHB15 and SHB20, the operator discretized gradient connecting the tensor of deformationto the vector of nodal displacements is given by:

. S

u Bd (27)

where:

,

,

,

, ,

, ,

, ,

u

u

u

u u

u u

u u

x x

y y

z z

Sx y y x

x z z x

y z z y

u ,

1

2

3

d

d d

d

(28)

and takes the practical matric shape then:




c2b0aa7ea6ee

This writing of the operator discretized gradient using the formulas of Hallquist [4] is very convenient

because the vectors

γ , which intervenes in the expression of B , check the following conditions of

orthogonality:

. 0

Tjγ x , .

Tγ h (30)

This makes it possible to separately handle each mode of the deformation to simply obtain the shapeof the field of applied deformation. Let us note that an element based on the formulation (29) is

convergent when it is evaluated exactly. However, the evaluation of this operator B , given in (29), of

each point of integration makes this element expensive in computing times for the practicalapplications, and the simplified shape of this element is essential.

3.3 Variational formulation used for elements SHB15 and SHB20

The extension of the weak form of the variational principle of Hu-Washizu to the case of the mechanicsof the nonlinear solids is due to Fish and Belytschko [6]. For a simple element, one a:

u , , =∫V e

⋅ dV∫V e

⋅∇ s u− dV− dT⋅ f ext=0 (31)

where represent a variation, u the field of displacement, is the rate of applied deformation, σ

the applied constraint, σ the constraint evaluated by the constitutive law, d nodal displacements, extfexternal nodal forces, and ∇ s u the symmetrical part of the gradient of the field of displacement.

The formulation “ Assumed strain ” (projection of the operator discretized gradient B on under suitable

space in order to avoiding the various problems of blocking) is based on a simplified form of thevariational principle of Hu-Washizu as it was described by Simo and Hughes [7]. In this simplified form,the applied constraint is selected orthogonal with the difference between the symmetrical part of thegradient of displacement and the rate of applied deformation. Thus, the second term in the equation(31) be eliminated and one obtains:

u , , =∫V e

⋅ dV− dT⋅f ext=0 (32)

In this form, the variational principle is independent of the interpolation of the constraint, since theapplied constraint does not intervene any more and thus need does not have to be defined. The

discretized equations thus require the only interpolation of displacement u and of the rate of applied

deformation in the element. With the preceding vectorial notations one a:

u x ,t =∑i=1

15

d I t N I x (element SHB15)




c2b0aa7ea6ee

Or u x ,t =∑i=1

20

d I t N I x (element SHB20) (33)

This led to:

∇ s u x , t =B xd t (34)

The applied deformation is defined as for it by:

x , t = B x d t (35)

Replacing the expression (35) in the variational principle (32), one obtains:

dT ∫V e

B⋅ dV−F ext =0 (36)

Like d can be arbitrarily selected, the preceding equation leads to:

int extf f (37)

with:

f int=∫V e

B x ⋅ dV (38)

In the equation above, it is well specified that the constraint σ is calculated by the law constitutive

starting from the rate of applied deformation. For the nonlinear problems, σ can also be an integral

function of the rate of applied deformation and other internal variables:

=F , , ... (39)

where α represent the internal variables. The formulation thus obtained is valid for problems including

the two types of nonlinearities: geometrical and material. In the case of linear problems, one a:

. . . σ Cε CBd (40)

The elastic matrix of behaviorC , in the case of an isotropic material, is selected like following:

2 2

2 2

0 0 0 01 1

0 0 0 01 1

0 0 0 0 0

0 0 0 0 02 1

0 0 0 0 02 1

0 0 0 0 02 1

E E

E E

E

E

E

E

C

In this matrix, E is the modulus Young and ν is the Poisson's ratio. This law is specific to elementsSHB. It resembles that which one would have in the case of the assumption of the plane constraints,




c2b0aa7ea6ee

put except for the term (3.3). We can note that this choice involves an artificial anisotropic behavior.This choice makes it possible to satisfy all the tests without introducing blocking.

The internal forces of the element are written then simply in terms of the elementary matrix of rigidity:

f int=K e⋅d (41)

where:

K e=∫V e

BT⋅C⋅B dV (42)

In a standard approach in displacement, the rate of applied deformation is identified with thesymmetrical part of the gradient speed, which amounts replacing B by B in the preceding expressions.One thus obtains simply:

K e=∫V e

BT⋅C⋅B dV (43)

The prismatic element with 15 nodes, named “SHB15”, has 15 points of integration. Their coordinates

, , and their weights of integration are the roots of the polynomial of Gauss-Legendre given in

the following table:

Thus, the expression of rigidity is: K e=∑j=1

15

w P j J P j BT P j ⋅C⋅BP j

The coordinates of the points of Gauss and their weights for element SHB20 are given in the tablebelow:


NotGauss w , ,

P (1) 1/2 1/2 1 0.906179845938664G 5056189/60.23692688

P (2) 1/2 1/2 2 0.538469310105683G 499366/6.4786286700

P (3) 1/2 1/2 3 0G 8888889/60.56888888

P (4) 1/2 1/2 4 0.538469310105683G 499366/6.4786286700

P (5) 1/2 1/2 5 0.906179845938664G 5056189/60.23692688

P (6) 0 1/2 6 0.906179845938664G 5056189/60.23692688

P (7) 0 1/27 0.538469310105683G 499366/6.4786286700

P (8) 0 1/28 0G 8888889/60.56888888

P (9) 0 1/29 0.538469310105683G 499366/6.4786286700

P (10) 0 1/210 0.906179845938664G 5056189/60.23692688

P (11) 1/2 011 0.906179845938664G 5056189/60.23692688

P (12) 1/2 012 0.538469310105683G 499366/6.4786286700

P (13) 1/2 013 0G 8888889/60.56888888

P (14) 1/2 014 0.538469310105683G 499366/6.4786286700

P (15) 1/2 015 0.906179845938664G 5056189/60.23692688



c2b0aa7ea6ee

Thus, the expression of rigidity is: K e=∑j=1

20

w P j J P j BT P j ⋅C⋅BP j

3.4 Geometrical matrix of rigidity Ksigma

The matrix K aims to solve the problems of buckling. We point out here that the modes of bucklingare the clean vectors of the problem to the eigenvalues generalized according to:

K K ⋅u=0⇔ K⋅u= K⋅u

with , and is the multiplying coefficient of the loading.

By introducing the quadratic deformation Qe such as:

e ijQu , u=∑

k=1

3

uk ,i . uk , j


NotGauss w , ,

P (1) 1 3 1 3 1 0.906179845938664G 0.236926885056189

P (2) 1 3 1 3 2 0.538469310105683G 0.478628670499366

P (3) 1 3 1 3 3 0G 0.568888888888889

P (4) 1 3 1 3 4 0.538469310105683G 0.478628670499366

P (5) 1 3 1 3 5 0.906179845938664G 0.236926885056189

P (6) 1 3 1 3 6 0.906179845938664G 0.236926885056189

P (7) 1 3 1 3 7 0.538469310105683G 0.478628670499366

P (8) 1 3 1 3 8 0G 0.568888888888889

P (9) 1 3 1 3 9 0.538469310105683G 0.478628670499366

P (10) 1 3 1 3 10 0.906179845938664G 0.236926885056189

P (11) 1 3 1 3 11 0.906179845938664G 0.236926885056189

P (12) 1 3 1 3 12 0.538469310105683G 0.478628670499366

P (13) 1 3 1 3 13 0G 0.568888888888889

P (14) 1 3 1 3 14 0.538469310105683G 0.478628670499366

P (15) 1 3 1 3 15 0.906179845938664G 0.236926885056189

P (16) 1 3 1 3 16 0.906179845938664G 0.236926885056189

P (17) 1 3 1 3 17 0.538469310105683G 0.478628670499366

P (18) 1 3 1 3 18 0G 0.568888888888889

P (19) 1 3 1 3 19 0.538469310105683G 0.478628670499366

P (20) 1 3 1 3 20 0.906179845938664G 0.236926885056189



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One can define this matrix of geometrical rigidity by:

uT⋅K⋅ u=∫0

: eQ u , ud=∫ 0

:∇uT∇ u d

In order to express this matrix in discretized space, let us introduce the discretized operators quadratic

gradient QB (in matric notation) such as:

11

11

2222

3333

12 21 12

13 3113

23 32

23

T Q

Q

T QQ

T QQQ

Q Q T Q

Q QT Q

Q Q

T Q

e

e

eδ ,

e e

e e

e e

u B u

u B u

u B ue u u

u B u

u B u

u B u

Various terms Q

ijB are given by the following equations:

1 1

1 111

1 1

Q

T

T

T

B B 0 0

B 0 B B 0

0 0 B B

;

2 2

2 222

2 2

Q

T

T

T

B B 0 0

B 0 B B 0

0 0 B B

;

3 3

3 333

3 3

Q

T

T

T

B B 0 0

B 0 B B 0

0 0 B B

1 2 2 1

1 2 2 112

1 2 2 1

Q

T T

T T

T T

B B B B 0 0

B 0 B B B B 0

0 0 B B B B

1 3 3 1

21 3 3 113

1 3 3 1

Q c

T T

T T

T T

B B B B 0 0

B 0 B B B B 0

0 0 B B B B

2 3 3 2

22 3 3 223

2 3 3 2

Q c

T T

T T

T T

B B B B 0 0

B 0 B B B B 0

0 0 B B B B

with the vectors iB ( I = 1,2,3) defined as:

,i i ih B bγ




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With these notations, the contribution to the geometrical matrix of rigidity, k , at the point of Gauss

j is given by:

11 22 3311 22 33

12 13 2312 13 23

Q Q Qj j j j j j j

Q Q Qj j j j j j

k B B B

B B B

By integration on the points of Gauss of the element, the geometrical matrix of rigidity is obtained bythe formula:

K =∑j=1

5

w j J jk j for element SHB15 and element SHB20

3.5 Following forces and matrix of pressure Kp

The following compressive forces are present in the tangent matrix via the matrix K p , because thefollowing external forces depend on displacement. The following compressive forces are written:

∫∂pnT⋅udS=∫∂ 0

p det [F u]n0T F u -TdS 0= p F0− p K p⋅u

F u=1∇ u by using the notations:

• 0 1 2 3( )T n n n n , normal on the surface external of the element in the configuration of

reference;• bi , vector of size 6 (for SHB15) or 8 (for SHB20), derived from the functions of form to the 6

(for SHB15) or 8 (for SHB20) nodes of the face of the element charged in pressure;

• 0S surface of the face charged in pressure.

The preceding formulation leads to a not-symmetrical matrix. It is known that one can nevertheless usea symmetrical formulation if the external forces due to the pressure derive from a potential. It is thecase if the compressive forces do not work on the border of the modelled field. It is thus consideredthat the symmetrical part of the matrix is enough. The symmetrized matrix takes the following shape:




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It is a matrix 18×18 or 24×24 , that it is necessary to multiply by displacements of the 6 (forSHB15) or 8 (for SHB20) nodes of the face to which one applies a pressure.




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4 Strategy for non-linear calculations

4.1 Geometrical non-linearities

One treats here the case of great displacements, but weak rotations and small deformations. One adopts forthat an up to date put Lagrangian formulation.

Into nonlinear, we seek to write balance between internal forces and force external at the end of theincrement of load (located by index 2):

extFF 2int2

The expression of the internal forces is written:

2

int2 22

.T dV

F Bσ

In the preceding equation the operator 2B is the operator allowing to pass from the displacement to the

linear deformation calculated on the geometry at the end of the step, the constraint 2σ is the constraint of

Cauchy at the end of the step and integration is made on volume 2 deformed at the end of the step.

Important remarks:

• For the element SHB6, the matrix 2

B is also modified by “ Local Assumed strain method “. It

takes the following shape:

• For elements SHB15 or SHB20, the matrix 2

B need for modification does not have. It thus takes,

the following form:




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The element available to date in Aster is programmed in small rotations. Indeed, the increment of deformation iscalculated by using only the linear deformation:

E=12∇ 1 u∇1

T u

The operator gradient is calculated on the geometry of beginning of step. This writing of the deformation islimited to small rotations (lower than 5 degrees).

One could without difficulty of extending the formulation to great rotations by including in the deformation theterms of second order (tensor of Green-Lagrange):

E= 12∇1 u∇1

T u∇1

T u⋅∇ 1 u

The associated tensor of constraint is the second tensor of Piola Kirchhoff II [R5.03.22]. But this is not availablein version 12 of Code_Aster.

In elasticity, the law of behavior is written:

C=C ' E

where C ' is the matrix of Hooke. Let us notice that for the elements SHB, this matrix is a transverseorthotropic matrix which is written in the axes of the lamina:

2 0 0 0 0

2 0 0 0 0

0 0 0 0 0'

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

E

C

4.2 Non-linearities materials

Into non-linear materials, we propose a method of particular construction of the tangent matrix CT . It consistsin supposing initially that the element is in a state of plane constraint in the local reference mark of each point ofintegration of Gauss and the deformations except plan are elastic. That involves then immediately that the totaldeflections except plan are equal to the elastic strain. Let us call CCPT the tangent matrix in plane constraints.The tangent matrix of behavior for the selected behavior and is written:




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0 0 0

0 0 0

0 0 0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

CPT CPT CPTxxxx xxyy xxxy

CPT CPT CPTxyyx yyyy yyxy

T

CPT CPT CPTxyxx xyyy xyxy

C C C

C C C

EC

C C C

Then the constraints except plan are calculated in an elastic way. This method thus makes it possible to

connect the elements SHB with all the laws of behavior available in Code Aster.




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5 Establishment of elements SHB in Code_Aster

5.1 Description

These elements are pressed on the voluminal meshs 3D PENTA6, HEXA8, PENTA15 and HEXA20.

5.2 Use

These elements are used in the following way:

5.2.1 Grid

To check the good orientation of the faces of the elements indicated (compatibility with the privilegeddirection) while using ORIE_SHB of the operator MODI_MAILLAGE.

5.2.2 Modeling

The name of modeling SHB was preserved. It is of course an abuse language, this modeling nowgathering the 4 finite elements SHB6, SHB8, SHB15, SHB20.

5.2.3 Material

For a homogeneous isotropic elastic behavior in the thickness one uses the keyword ELAS inDEFI_MATERIAU where the coefficients are defined E, Young modulus and NAKED, Poisson's ratio.

To define a plastic behavior the keyword is used TRACTION in DEFI_MATERIAU where one defines thename of a traction diagram. Only this kind of definition is available for the moment.

It should be noted that thermal dilation is not taken into account in version 12 of Code_Aster forelements SHB.

5.2.4 Boundary conditions and loading

One imposes the boundary conditions on the degrees of freedom of volume 3D (AFFE_CHAR_MECA /DDL_IMPO), and efforts in the total reference mark (FORCE_NODALE).

One defines the efforts of pressure distributed on the faces of the element (under the keywordPRES_REP). One will have taken first care to define meshs of skin QUAD4 and to suitably direct theoutgoing normals with these meshs of skin using the orderMODI_MAILLAGE keyword ORIE_PEAU_3D

5.2.5 Calculation in linear elasticity

Order MECA_STATIQUEThe options of postprocessing available are SIEF_ELNO and SIEQ_ELNO.

5.2.6 Calculation in linear buckling

The option RIGI_MECA_GE being activated in the catalogue of the element, it is possible to carry out aclassical calculation of buckling after assembly of the matrices of elastic and geometrical rigidity.

5.2.7 Calculation in geometrical nonlinear “elasticity”




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The behavior is chosen ELAS under the keyword BEHAVIOR of STAT_NON_LINE, in smalldeformations (‘SMALL‘) or in great displacements and small rotations (‘GROT_GDEP’) under thekeyword DEFORMATION. In this last case, only the geometry is brought up to date at the beginning ofstep of time, the behavior remains calculated in small deformations.

The strategy used being based on the use of a matrix of tangent rigidity during iterations(reactualization at the beginning of step only), one will take care to use another option only that whichis activated by default, namely REAC_ITER = 0 under NEWTON.

Digital integration in the thickness is carried out with 5 points of Gauss, just like in nonlinear material.

5.2.8 Calculation nonlinear plastic

Only the criterion of Von Mises is available to date (RELATION = ‘VMIS_ISOT_TRAC‘ underBEHAVIOR). One defines the way of calculating of the deformations as in the case of nonlinearelasticity (DEFORMATION = ‘GROT_GDEP‘ or ‘SMALL‘).

The strategy used being based on the use of a matrix of tangent rigidity during iterations(reactualization at the beginning of step only), one will take care to use another option only that whichis activated by default, namely REAC_ITER = 0 under NEWTON.

5.3 Establishment

Options RIGI_MECA, RIGI_MECA_GE, FORC_NODA, FULL_MECA, RIGI_MECA_TANG, RAPH_MECA,SIEF_ELGA, SIEF_ELNO were activated in the catalogue gener_shb3d_3.catastrophes.

No development was necessary for the compressive forces distributed and the following compressiveforces. Indeed, these loadings are pressed on meshs of skin identical to those of the voluminalelements 3D.

5.4 Validation

The tests validating these elements are:

Tests into linear:

• SSLS101 C, D, K, L: circular plate simply posed subjected to a uniform pressure [V3.03.101]Modeling C: SHB8, Modeling D: SHB20, Modeling K: SHB6, Modeling L: SHB15.

• SSLS105 C: hemisphere doubly pinch [V3.03.105] classical test to check the convergence of element (SHB8)

• SSLS108 C with H: beam bored in inflection, test allowing to check the absence of blocking [V3.03.108]Modelings C, D: SHB8, Modeling G: SHB20, Modelings E, F: SHB6, Modeling H: SHB15.

• SSLS123 a: sphere under external pressure [V3.03.123] to validate the loadings of pressure and the orthotropic behavior particular to this elementModeling A: SHB8, Modelings C, D: SHB6.

• SSLS124 A with G: thin section in inflection with various twinges, to delimit the field of use of the element [V3.03.124]. Modelings A, B: SHB8, Modeling C: SHB6, Modelings D, E: SHB20, Modelings F, G: SHB15.

• SSLS125 A, b: buckling (modes of Euler) of a free cylinder under external pressure [V3.03.125] this testmakes it possible to validate the geometrical nature of rigidityModeling A: SHB8, Modeling b: SHB20.




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Tests into nonlinear:• SSNS101 C with G: breakdown of a cylindrical roof [V6.03.101]. This test makes it possible to validate geometrical nonlinear calculation and elastoplasticityModelings C, D: SHB8, Modeling E: SHB20, Modeling F: SHB6, Modeling G: SHB15.

• SSNS102 A, b: buckling of a hull with stiffeners in great displacements and following pressure [V6.03.102].Modeling A: SHB8, Modeling b: SHB20.

6 Bibliography

[1] F. Abed-Meraim and A. Combescure. SHB8PS, has new adaptive, assumed strain continuummechanics Shell element for impact analysis. Computers and Structures , 80:791-803, 2002.

[2] F. Abed-Meraim and A. Combescure. Stabilization of the under-integrated finite elements.Report interns n° 247, LMT of Cachan, 2001.

[3] A. Legay. An effective method of calculating for parametric study of non-linear buckling of thethree-dimensional structures: application to reliability. Doctorate of the LMT of Cachan, 2002.

[4] T. Belytschko, J.S. - J. Ong, W.K. Liu, and J.M. Kennedy. Hourglass control in linear andnonlinear problems. Comput. Methods Appl. Mech. Eng., 43:251 - 276, 1984.

[5] NR. Caironi and F. Abed-Meraim. Implementation of element SHB6 to integration reduced inthe INCA code and analyzes instabilities of the hourglass type. Report of project of end ofstudies at laboratory LPMM, ENSAM Metz, June 2003.

Description of the versions of the document:

Version Aster

Author (S), Organization (S) Description ofthemodifications

9.5 Trinh Vuong Dieu (thesis) X Desroches EDF R & D AMA Initial version


elements of voluminal hull shb with 6.15 and 20 noeu []

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