code-aster.orgcode-aster.org/V2/doc/default/en/man_r/r7/r7.02.12.pdfCode_Aster Version default Titre : eXtended Finite Element Method Date : 18/01/2017 Page : 5/114 Responsable : GÉNIAUT

Code_Aster Versiondefault

Titre : eXtended Finite Element Method Date : 05/12/2017 Page : 1/115Responsable : GÉNIAUT Samuel Clé : R7.02.12 Révision :

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eXtended Finite Method Element: General informationS

Summary:

This document presents the method X-FEM (eXtended Finite Method Element) which mainly makes it possibleto consider cracks not respecting the grid to deal with the problems of cracks 2D and 3D. The crack is definedby the order DEFI_FISS_XFEM [U4.82.08] and is usable for calculations in linear and non-linear statics.

Other documents dedicated to specific problems are available: • contact-friction on the lips of the crack in small slips with X-FEM [R5.03.54], • contact-friction on the lips of the crack in great slips with X-FEM [R5.03.53] , • propagation of cracks with X-FEM [R7.02.13]

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 2018 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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Contents

1Introduction............................................................................................................................................ 5

2Representation of the crack by “level sets”............................................................................................7

2.1Theoretical aspect of the level sets.................................................................................................7

2.2Calculation of the level sets............................................................................................................9

2.2.1Calculation of the level set normal (lsn).................................................................................9

2.2.1.1Calculation of the level set normal in 3D....................................................................9

2.2.1.2Calculation of the level set normal in 2D..................................................................11

2.2.2Calculation of the level set tangent (lst)...............................................................................13

2.2.3Approximation of the level sets............................................................................................15

2.2.4Readjustment of the level sets.............................................................................................18

2.2.5Concept of true signed distance..........................................................................................20

2.2.6Multi-cracking....................................................................................................................... 21

2.3Base local at the bottom of crack..................................................................................................21

2.4Determination of the bottom of crack............................................................................................22

2.4.1Research of the points of the bottom of crack......................................................................22

2.4.2Orientation of the bottom of crack........................................................................................24

2.4.2.1Explanation of the method of orientation of the bottom of crack..............................24

2.4.2.2Case of a multiple bottom of crack...........................................................................26

2.4.2.3Algorithm used for the orientation of the bottom......................................................27

3Problem of cracking with X-FEM.........................................................................................................30

3.1Problem general............................................................................................................................ 30

3.2Enrichment of the approximation of displacement.........................................................................31

3.2.1EnrichesmeNT with a function of selection of field (2ème term)..........................................31

3.2.2Enrichment with a function of selection of field for a connection of cracks...........................32

3.2.3Enrichment with the singular functions (3ème term)............................................................37

3.2.4Geometrical enrichment.......................................................................................................41

3.2.5Enrichment in Code_Aster...................................................................................................42

3.2.5.1Enrichment (statute) of the nodes............................................................................42

3.2.5.2Enrichment (statute) of the meshs...........................................................................44

3.2.5.3Cancellation of the degrees of freedom nouveau riches “in excess”........................45

3.2.6Conditioning related to enrichment......................................................................................45

3.2.6.1General information on conditioning XFEM.............................................................45

3.2.6.2Estimated criteria to improve conditioning...............................................................46

3.2.6.3Pre-conditioner XFEM for the matrices....................................................................48

3.3Under-cutting................................................................................................................................ 50

3.3.1Preliminary phase of cutting of the hexahedrons to bring back itself to tetrahedrons..........51

3.3.2Preliminary phase of cutting of the pentahedrons to bring back itself to tetrahedrons.........53




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3.3.3Preliminary phase of cutting of the pyramids to bring back itself to tetrahedrons................54

3.3.4Under-cutting of a tetrahedron under-tetrahedrons..............................................................55

3.3.4.1Three points of intersection of which two points tops...............................................56

3.3.4.2Three points of intersection of which a point top......................................................58

3.3.4.3Five points of intersection including two points tops................................................58

3.3.4.4Three points of intersection of which no point top....................................................59

3.3.4.5Four points of intersection of which a point top........................................................60

3.3.4.6Six points of intersection including two points top....................................................61

3.3.4.7Four points of intersection.......................................................................................61

3.3.4.8Five points of intersection of which a point top........................................................62

3.3.4.9Four points of intersection of which a point top........................................................62

3.3.5Multi-cutting......................................................................................................................... 63

3.3.6Maximum number of subelements.......................................................................................64

3.3.6.1Case of the tetrahedron...........................................................................................64

3.3.6.2Case of the pentahedron.........................................................................................64

3.3.6.3Case of the hexahedron..........................................................................................65

3.3.6.4Case of multi-cutting................................................................................................65

3.3.7Under-cutting 2D..................................................................................................................65

3.3.8Algorithms of under-cutting..................................................................................................67

3.3.8.1Calculation of the points of intersection...................................................................67

3.3.8.2Calculation of the points medium.............................................................................72

3.4Recovery of the facets of contact..................................................................................................82

3.4.1Elements XH........................................................................................................................82

3.4.2Elements XHT and XT.........................................................................................................84

3.4.3The multi-Heaviside elements..............................................................................................84

3.4.4Maximum number of recovered facets.................................................................................87

3.4.4.1Case 2D...................................................................................................................87

3.4.4.2Case 3D...................................................................................................................87

3.4.4.3Case multi-Heaviside...............................................................................................89

3.5Integration rigidity.......................................................................................................................... 89

3.5.1Intégrande of the term of rigidity mechanics........................................................................89

3.5.2Intégrande of the geometrical term of rigidity.......................................................................89

3.5.3Calculation of the derivative of the singular functions:.........................................................90

3.5.4Diagrams of integration........................................................................................................92

3.5.4.1Integration of the nonsingular terms........................................................................92

3.5.4.2Integration of the singular terms..............................................................................92

3.6Integration of the second surface members..................................................................................93

3.6.1Intégrande of the second member of the surface efforts......................................................93

3.6.2Intégrande of the second member of the surface efforts on the lips of the crack.................93




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4Linear thermics with X-FEM.................................................................................................................96

4.1Introduction................................................................................................................................... 96

4.2Restrictions................................................................................................................................... 96

4.3With dealt problem........................................................................................................................96

4.4Approximation X-FEM of the field of temperature.........................................................................98

4.5Integration of the matrices and the elementary vectors.................................................................99

4.5.1Voluminal integrals...............................................................................................................99

4.5.2Surface integrals................................................................................................................100

5Energy calculation of Factors of Intensity of the Constraints.............................................................101

5.1Method G-theta for calculation of G............................................................................................101

5.1.1Relations of balance..........................................................................................................101

5.1.2Lagrangian expression of the rate of refund of energy.......................................................102

5.1.3Discretizations................................................................................................................... 104

5.2Method G-theta for calculation of KI, KII and KIII with the level sets...........................................104

5.2.1Bilinear form of G...............................................................................................................105

5.2.1.1Expression of S1 and S1TH...................................................................................105

5.2.1.2Expression of S2 and S2TH...................................................................................106

5.2.1.3Surface term..........................................................................................................107

5.2.1.4Thermal term.........................................................................................................107

5.2.2Separation of the mixed modes.........................................................................................108

5.2.3Discretizations................................................................................................................... 109

5.3Method G-theta with X-FEM........................................................................................................110

6Bibliography....................................................................................................................................... 111

7Description of the versions................................................................................................................. 115




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1 Introduction

The digital simulation in breaking process is primarily based on the use of the finite element method(MEF). Classically for a problem of propagation of cracks, the first calculation is carried out on an initialgrid, then a new grid is given, fascinating of account the projection of the crack (according to the law ofselected propagation). A new calculation is resulted from this, and the process is reiterated for eachstep of propagation.

A major drawback and immediate is the need for re-meshing with each step of propagation. Theprocess of mending of meshes can be easily automated in 2D [feeding-bottle1], and in certain cases in3D [feeding-bottle2], but a mending of meshes 3D of quality proves to be expensive in time (humansupervision) and money. Indeed with an automatic maillor, a local refinement appropriate to the level ofthe zone of cracking often involves an excessive number of elements everywhere on the rest of thestructure. A process of refinement by layers is generally useful, like the introduction of a torus in bottomof crack, powerful solution (also facilitating the installation of elements of Barsoum [feeding-bottle3])but requiring an expensive human intervention; and this more especially as the geometrical shape ofthe crack is very complex (helicoid cracks for example [feeding-bottle4]). The problem becomes quasi-inconceivable in multi-cracking 3D. Besides these practical difficulties, the projection of sizes (forced,internal variables) from one grid to another poses fundamental theoretical problems (checking of theconservation equations of energy, momentum, mass) [feeding-bottle5].

Parallel to the difficulties related to the propagation, the methods with grid prove not very effective forparametric studies where one is interested in the influence of the position and the fissure shape.

The methods known as “Meshless” [feeding-bottle6] were proposed to free itself from the constraintsrelated to the grid. The meshless are based on a discretization only nodal, without connectivity, and thefunctions of form are built starting from the nodal configuration. Initially introduced at the end of theSeventies ' for the problems without borders (method Smoothed Particle Hydrodynamics in the field ofastrophysics), the Meshless methods were wide thereafter with the mechanical problems and aredeclined today under various alternatives: methods MLS (Moving Least Public garden), DiffusesElement Method (DEM), Element Free Galerkin (EFG), Reproducing Kernel Particle Method (RKPM),HP clouds, and well of others. The common element of all these methods is the concept of Partition ofthe Unit, which is a set of functions whose sum is equal to one in each point of the field considered.The principal disadvantage of these methods is that they require a digital effort greater than thosebased on a grid (time CPU in particular). In particular, the evaluation of the functions of form is far frombeing also commonplace, the digital diagrams of integration are generally richer thus more expensiveand the total system of equations resulting has a higher bandwidth compared to a MEF [feeding-bottle7]. Even in the recent versions of combining the Meshless methods of the physical andmathematical supports [feeding-bottle8], the imposition of boundary conditions remains problematic.

Other methods are based on a partition of the unit within the framework of the standard finite elements,where only the definition of the functions of forms differs. This choice of partition of the unit avoids theproblems of integration of rigidity, considerably expensive for the methods Meshless (EFG, DEM,RKPM in particular). Moreover, the use of a partition of the unit based on the finite elements allows aneasy implementation of the boundary conditions of the Dirichlet type (contrary to the techniques using apartition of the unit based on least squares). These methods, which one initially finds in their versionswith a grid under the name of Partition of Unity Finite Method Element (PUFEM) or of GeneralizedFinite Element Method (GFEM) [feeding-bottle9], allow to easily enrich space by the functions of form,thanks to knowledge a priori properties of the solution of the problem. Combining the GFEM and HP-cloud, Duarte et al. [feeding-bottle10] a partition of the mixed unit proposes (finite elements andShepard), which makes it possible to consider a crack nonwith a grid, with functions of enriched forms.

Still nearer to the classical framework finite elements, the finite element method wide (X-FEM) causedone of sharpest the interest if one refers about it to the evolution amongst publications on this subjectsince his appearance and in the place which is reserved to him in the international conferences. Thismethod uses the partition of the unit to enrich the base by the functions of form in order to represent ajump of the field of displacement on the level of the lips of the crack, as well as the singularity in bottom




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of crack. Two enrichments are then introduced: an enrichment by a function jump which makes itpossible to manage discontinuity through the lips of the crack, and an enrichment by asymptoticfunctions, which allows a representation faithful of the physical phenomena taking place to the level ofthe bottom of crack. Historically, the precursory signs are due to Belytschko and Black [feeding-bottle11], which applies the partition of the unit [feeding-bottle23] with the breaking process byincorporating the analytical formulas of the asymptotic fields in the approximation of displacement. Theaddition of the Heaviside function generalized [feeding-bottle24] allows to write the approximationenriched by displacement in its final form, expression which gives rise to the finite element methodwide. Compared with GFEM, it offers less dependence with respect to the knowledge of the form of thesolutions, which offers a greater flexibility [feeding-bottle15]. The field of application of X-FEM does notcease widening, this approach having been used within very varied frameworks: plates of Reissner-Mindlin, hulls rupture in 3D, multi-cracking, zones cohesive, modeling of holes and bi--materials,formulations incompressible and great transformations, nucleation of the cracks, cracking undercontact, dynamic rupture, plasticity…

Moreover, the use of the method of the level sets coupled with X-FEM largely facilitated the treatmentof the cracks in 2D ([feeding-bottle15], [feeding-bottle14] and [feeding-bottle17]) and in 3D [feeding-bottle18]. Initially introduced within the framework of the mechanics of the fluids to represent theevolution of interfaces, the method of the level sets regards the interface as the Iso-zero of a functionoutdistances. This method proves particularly effective for the propagation of a crack in 3D [feeding-bottle12], coupled with the use of Fast Marching Method [feeding-bottle16].

This document is articulated around 4 sections, of which this introduction which holds place of section5.The section 2 is devoted to the use of the method of the level sets for cracking. After a short theoreticalrecall, one specifies the calculation of the level sets, which prove to be practical to determine theposition of the crack in 3D and the bottom of crack, and which makes it possible to define a local basein bottom of crack.The section 3 present the problem of cracking treated with X-FEM. One introduces in the paragraph[§3.2] the approximation of displacement writes in a base of functions of enriched form. The addition ofdiscontinuous functions through the interface leads to a procedure of under-cutting detailed in theparagraph [§3.3], precondition to the phase of integration of the terms of rigidity and the secondmember, whose installation is explained in the paragraph [§3.5].

The section 5 be interested in postprocessing of the rate of refund of energy and the stress intensityfactors in linear breaking process. Method G - theta makes it possible to calculate the rate of refund ofenergy room. For that, a field theta is introduced, representing a virtual extension of the crack. Groom is then solution of a variational equation, using the integral J in the form of integrals of field.Choices of discretizations of G and of the field theta bring to a linear system, of which the resolutionleads to the values of G along the bottom of crack [feeding-bottle48]. The method G-theta also makesit possible to determine the stress intensity factors along the bottom of crack. Instead of using theintegral J , one uses the bilinear form of G , which leads to integrals of field mixing analytical fieldssolutions and asymptotic fields (called also integral of interaction). Thus, just as for G , them K arethen solutions of a variational equation, using integral of interaction. Choices of discretizations of Kand of the field theta bring to a linear system, of which the resolution leads to the values of K alongthe bottom of crack [feeding-bottle49].




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2 Representation of the crack by “level sets”

The method of the “Level sets” was introduced within the framework of the mechanics of the fluids torepresent the evolution of interfaces. The principal idea is to regard the interface as the Iso-zero of afunction outdistances. The choice of the function distance imports little here, because only theknowledge of the Iso-zero is useful and important.

2.1 Theoretical aspect of the level sets

That is to say an interface delimiting open of ℜn . The idea is to define a function x , t

regular (at least Lipchitzienne) such as the subspace x , t =0 represent the interface.The level set has the following properties:

x , t 0 pour x∈ x , t 0 pour x∉

x , t =0 pour x∈∂=Γ t .

This method applies easily to the problems of cracking 2D, in particular within the framework of theapproaches where the crack is not with a grid. ([feeding-bottle14] [feeding-bottle15] in 2D). Theextension is possible for the treatment of the cracks in 3D.

Thus, in the case of cracking, it is necessary to introduce two level sets ([feeding-bottle17] in 2D and[feeding-bottle18] in 3D):

• a level set normal ( lsn ) who represents the distance to the surface of the crack (surfaceextended by prolongation to all the field),

• a level set tangent ( lst ) who represents the distance to the bottom of crack.

Figure 2.1-1 : Level sets and distance to the crack

The Iso-zero of the level set normal defines the surface of the crack, extended by continuity to all thefield. The intersection of the Iso-zero of both level sets defines the bottom of crack. Moreover, the signof the level set tangent is selected so that the surface of the crack Γ cr corresponds to the spacegenerated by lsn=0 ∩ lst0 . The sign of the level set normal is arbitrarily selected thanks to theconvention of orientation of the normal to the plan of crack, explicitly definite in the paragraph [§2.2.1].Points x for which lsn x is negative are known as “below” the crack, and those for which lsn x is positive are known as “above” the crack (see Figure 2.1-2 and Figure 2.1-3).




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Figure 2.1-2 : Level sets for the representation of a crack 3D

Figure 2.1-3 : Level sets for the representation of a crack 2D

The phase of propagation of the crack results simply in the propagation of the level sets. Thepropagation of a level set requires three successive stages [feeding-bottle13]:

•extension speed known on the Iso-zero towards the whole field,•propagation of the level set starting from this field speed,•rebootstrapping of the function level set in order to preserve a function outdistances signed.

propagation of a crack represented by 2 level sets presents some characteristics. The Flight PathVector is known only on the face of crack, i.e. a curve. The crack can only grow, not to move. Twofunctions level sets must be propagated and it is wished that their gradients remain orthogonal. Thesequence of the stages can be to summarize like this:

•propagation of lsn and rebootstrapping of lsn , •propagation of lst , •orthogonalisation of the gradient of lst versus gradient of lsn , •rebootstrapping of lst .

These stages are reduced all to the solution of equation of the type Halimton-Jacobi [feeding-bottle12]:∂

∂ tF∣∇∣= f

where F and f are functions which depend on the stage.

Fast Marching Method [feeding-bottle16] is an alternative technique adapted well to the strictlymonotonous propagation of faces. This method separates the nodes from the grid following theirdistance of the interface, and with each iteration the equation of propagation is solved for the onlyimmediately adjacent nodes with the interface, by using a diagram of finished differences in order 2.

This part is more detailed in the doculies [R7.02.13].




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2.2 Calculation of the level sets

The calculation of the level sets (scalar fields) is carried out for each crack. It can be done in twomanners. Maybe by the data of their analytical expressions and in this case, a simple evaluation ofthese functions to the nodes of the grid provides the required scalar fields. That is to say the crack iswith a grid and in this case, it is necessary to give to surface meshs correspondents to a lip(GROUP_MA_FISS) and linear meshs correspondents at the bottom of crack (GROUP_MA_FOND). ). Inthe case 2D, one will give linear meshs (for GROUP_MA_FISS) and of the meshs points (forGROUP_MA_FOND). The distances are then calculated by an algorithm of orthogonal projectionsinspired by that used for the contact [feeding-bottle19], clarified in the paragraphs [§2.2.1] and [§2.2.2].

2.2.1 Calculation of the level set normal (lsn)

For each node of the grid, one seeks the mesh of GROUP_MA_FISS nearest to this node. For that, oneuses the algorithms of projection on a triangle (see the paragraph [§2.3.2] of [feeding-bottle19]) and ona quadrangle (see the paragraph [§2.3.3] of [feeding-bottle19]). The value of lsn is then the normaldistance from this point to the mesh.

2.2.1.1 Calculation of the level set normal in 3D

In 3D, it is necessary to pay attention within the meaning of the normal. Indeed, meshs ofGROUP_MA_FISS being interior with the structure, they are not meshs of edge, and the automaticorientation of the normals (keyword ORIE_PEAU_3D of the operator MODI_MAILLAGE) is thenimpossible. To ensure itself always to choose the same direction for the normals, one takes the normalof the first mesh of GROUP_MA_FISS as reference and one “propagates” the direction of this normalto all the other meshs of the group for each adjacent element. For each mesh one stores one indicatorof orientation mesh which is worth “+1” if the orientation of its normal is coherent with that of the meshof gradually built reference and “-1” if not. That makes it possible to affect the good sign of the value ofthe level set normal by multiplying its value not corrected by the indicator of orientation (see calculationalgorithm of lsn in 3D below).

Calculation algorithm of the indicator of orientation mesh:




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• creation of the vector I “indicating of orientation” and the vector C containing the number oflayer to which each element of GROUP_MA_FISS belongs. The size of the two vectors is equalto the number of elements n of GROUP_MA_FISS

• one puts at the zero all elements of the two vectors I and C , except the first element to whichone affects value +1

• buckle i on the number of the layer (vector C ), of 1 with n• buckle j on the elements of the vector C , of 1 with n

- one recovers the number of the layer to which the element j belongs:ncouche=C ( j)

- if ncouche=i , i.e. the element j belongs to the current layer i :• the normal is calculated n j with the element j (one calculates the

normal with the triangle formed by the first three nodes defining theelement)

• buckle nel on the elements which in common have at least a node withthe element j

• if the element nel does not belong to any layer ( C (nel)=0 ), oneassigns the number of the courante+1 layer to him ( C (nel)=i+1), one calculates the normal of it nnel and the indicator of

orientation: I (nel )=sign ( I ( j)⋅nnel⋅n j )• fine buckles

- end if• fine buckles

• fine buckles

Calculation algorithm of lsn in 3D:

•buckle on all the nodes P grid initialization of dmin

•buckle on the triangular meshs of GROUP_MA_FISS (with subdivision of the quadranglesin triangles)

are A , B and C tops of the trianglecalculation of the normal to the mesh: N=AB∧ACcalculation of M , project of P in the plan ABCif M apart from the triangle ABC , one brings back M on one of the right-handsides AB AC BC if M still apart from the triangle ABC , one brings back M in A B or Cit is recoveredindicator of orientation I normal of the meshif PMdmin then dmin=PM and lsn P =I⋅PM⋅N

•fine buckles

•fine buckles

The subdivision of a quadrangle in triangles makes it possible to be brought back to an approximatecalculation of projection in a linear case (because the functions of form of the quadrangle are bilinearwhereas those of the triangle are linear). Moreover, there exist 2 ways of carrying out such a cutting(according to the selected diagonal). All the possible cases are thus generated, as it is made in theparagraph [§2.3.3] of [feeding-bottle19].




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The calculation of projection within the framework of the search for pairing of the nodes of contact forthe method continues does not use this kind of algorithm, but solves the non-linear problem ofprojection on a quadrangle by the method of Newton. One could be inspired some for calculation bythe level sets by projection.

2.2.1.2 Calculation of the level set normal in 2D

In 2D, one starts by reorganizing the meshs of GROUP_MA_FISS to have a series of contiguous meshs.Two vectors will be stored, the list of the ordered meshs and lists it of their orientation, which one willcall here LIMA and ORI .

Algorithm of sorting of GROUP_MA_FISS :

Initialization of the first mesh M, first mesh of GROUP_MA_FISS. One has LIMA 1=M and

ORI 1=1 . The Boolean one is initialized FINFIS who tests if one is at the end of the crack,FINFIS initialized with FALSE .

•buckle of research of the following mesh LIMA k as long as FINFIS=FALSEif ORI k−1=1 , calculation of N node 2 of LIMA k−1if ORI k−1=0 , calculation of N node 1 of LIMA k−1FINFIS=TRUE

•buckle on the meshs M of GROUP_MA_FISS, as long as FINFIS=TRUECalculation of node 1 and node 2 of M , N1 and N2if N1=N , ORI k =1 and LIMA k =M , FINFIS=FALSEif N2=N , ORI k =0 and LIMA k =M , FINFIS=FALSE(left loop when one finds a mesh following)

•fine buckles(left loop if one did not find of following mesh because one is at the end of the crack)

•end of loop

Figure 2.2.1.2-1 : Orientation of the following mesh

The first identified mesh being able to be unspecified any mesh inside the crack, one carries out a shiftof the found list of the full number of mesh NBMAF less the number of found meshs k in order tohave a list which finishes on the mesh of end of found crack. One thus obtains a vector withnonaffected components at the head which represent the not yet indexed meshs and a part filled onthe end of the vector corresponding to the meshs directed previously stored.




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•buckle of research of the preceding mesh LIMA k as long as FINFIS=FALSE ( k decreasing)if ORI k1=1 , calculation of N node 1 of LIMA k1if ORI k1=0 , calculation of N node 2 of LIMA k1FINFIS=TRUE

•buckle on the meshs M of GROUP_MA_FISS, as long as FINFIS=TRUECalculation of node 1 and node 2 of M , N1 and N2If N1=N , ORI k =0 and LIMA k =M , FINFIS=FALSEIf N2=N , ORI k =1 and LIMA k =M , FINFIS=FALSE(left loop when one finds a mesh preceding)

•fine buckles(left loop if one did not find of preceding mesh, because one is at the end of the crack)

•end of loop

Figure 2.2.1.2-2 : Orientation of the preceding mesh

Calculation algorithm of lsn in 2D:

•buckle on all the nodes P gridinitialization of dmin•buckle on the linear meshs of LIMA

are A , B tops of the segment if ORI=1 and B , A if ORI=0calculation of M , project of P on the line ABif M apart from the segment AB , one brings back M in A or B•if PMdmin

dmin=PM and

lsn=PM P . sign AB ,AP where M P is the project of P on AB (possibly apart from AB )if ORI=0 , lsn=−lsn

•end if•fine buckles

•fine buckles




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2.2.2 Calculation of the level set tangent (lst)

For each node of the grid, one seeks the mesh of GROUP_MA_FOND nearest to this node. For that, oneuses the algorithm of projection on a segment (see the paragraph [§2.3.1] of [feeding-bottle19]). Thevalue of lst is then the normal distance from this point to the segment.Just as previously, the determination of the normal to the segment is not obvious a priori. To calculateit, it is initially necessary to find the mesh surface of GROUP_MA_FISS who borders it.

Calculation algorithm of lst in 3D:

•buckle on all the nodes P grid

initialization of dmin

•buckle on the segments of GROUP_MA_FOND

that is to say A and B two ends of the segment

•buckle on all the meshs of GROUP_MA_FISS

•if A and B belong to this mesh thenthat is to say C a node of the mesh other than A and Bcalculation of the normal to this mesh (càd the normal with the plan of crack): N=AB∧ACcalculation of the normal at the bottom of crack in the plan of thecrack:

N '=AB∧N

Checking of the direction of N 'That is to say M the projection of P on AB If M is apart from [AB ] , one brings back it in A or BIf PMdmin then dmin=PM and lst P =PM⋅N '

•end if

•fine buckles

•fine buckles

•fine buckles

Notice :

During projection of P on the segments AB bottom of crack, it possible that all is projectedM find themselves out of segment considered. In this, case, the fact of bringing back M on

the edge is not a sufficient criterion to determine the segment nearest (see Figure 2.2.2-2 ). Oneproposes to choose the segment nearest as that where the project the least was folded back.With the notations above, if one calls M ’ the folded back point (not B on Figure 2.2.2-2 , thenone seeks the segment for which the angle =MPM ’ is smallest. It is noted that in a casewithout folding back (when the project falls into the segment), this angle is null. This additionaltest is introduced for the choice of the segment nearest.




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Figure 2.2.2-1 : Calculation of the level set tangent

Figure 2.2.2-2 : Additional criterion for the choice of the segment nearest

Calculation algorithm of lst in 2D:•buckle on all the nodes P grid

initialization of dmin

•buckle on meshs not A of GROUP_MA_FOND

•buckle on all the meshs of GROUP_MA_FISS

•if A belongs to this mesh thenthat is to say B the second node of the meshthat is to say M the projection of P on ABcalculation of such as AM=×ABif AMdmin then dmin=AM and lst=−×AB

•finsi

•fine buckles

•fine buckles•fine buckles




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The following pathological case is considered:

The ABC mesh sees successively its three nodes tops to be made cancel level-set tangent:

• because of make-to-vertex criterion for nodes With and B as their support is traversed (§2.2.4)• because of criteria specific to the quadratic elements for node C.

Once level-set tangent of all them nodes tops cancelled, one a:

lstmax=max {lst ABC }=0 (1)

Therefore, when the quantity is calculated d=lstm / lstmax , there is a problem.

This situation can to be frequent in 2D with a crack “segment” (FORM_FISS=' SEGMENT') and a gridcharacterized by a strong gradient of size of mesh (and this all the more if the grid is triangular and free).

To correct, hasvant to calculate d , it is looked at if lstmax is worthless.SI yes, and if the particular conditions are respected:

• Tous them nodes tops must check |lst|<ε ;• the level-sets must be calculated since the catalogue of forms geometrical of DEFI_FISS_XFEM in the

case 2D hasvec FORM_FISS=' SEGMENT'• for each edge of the current mesh, one compares the length of his orthogonal project on the segment

(which constitutes the crack) has the length of this segment. At least one of these projected must have alength comparable (relative criterion) with that of the segment. Of this manner one makes sure that themesh is “far” from the crack (has less than the grid is not extremely coarse)

Then, O N shorts-circuit the end of the current iteration of the loop on the edges (that of the block relating to theadjustments specific to quadratic). If not, one stops in fatal error.

2.2.3 Approximation of the level sets

Whatever the method of calculating used (by analytical functions or projection), the fields of the levelsets are interpolated by the linear functions of form used for the approximation of the field ofdisplacement [feeding-bottle17]:


Figure 2.2.2-3: Problem of tangent cancellation of the level-set



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lsn(x )=∑i

ϕi (x )lsni

lst ( x )=∑i

ϕi( x ) lst i

where i are the classical linear functions of form and lsni and lst i nodal values of the fields levelsets. On a tetrahedron with 4 nodes the surface of the crack will be thus a plan, but on a hexahedronwith 8 nodes, it could be slightly curved.

The error of discretization of the level sets is directly dependent in keeping with grid and with the curve of the level set. Let us examine the following example, bringing into play a crack curves in 2D:That is to say the level set ellipse of equation:

x2 y0 .5

2

=1

The distance to the ellipse is calculated of each of the 4 nodes of the quadrangle defined by

x , y ∈[0 .4,0 .9 ]×[ 0,0 .5 ] . That is to say P a given point of the space of cordonnées x p , y p .That is to say H its projection on the ellipse previously definite. H has as coordinates

a cosθ ,b sin θ . The equation of the right-hand side HP is the following one:

y=a sin θb cosθ

xb sin θ 1−a2

b2 This equation of unknown factor numerically is solved θ for each node of the quadrangle and onefrom of deduced the distance to the ellipse:

dist= x p−acos θ 2 y p−bsin θ

2

The approximation of the level set is written then on this quadrangle:

lsnh x , y =−0.443471 x , y −0.12 x , y 0.222273 x , y 0.0408064 x , y

where ϕi i=1,4 are the functions of form associated with the nodes with the quadrangle. Thesefunctions is expressed using N i , classical functions of form on the quadrangle of reference, and the

changes of variables between the real coordinates ( x , y ) and coordinates of reference ( s ,t ) .

ϕ1 ( x , y )=N 1 ( s (x , y) , t (x , y ))=14

( 1−s ) ( 1−t )

ϕ2 ( x , y )=N 2 ( s (x , y ) , t ( x , y))=14

( 1+ s) ( 1−t )

ϕ3 ( x , y )=N 3 ( s ( x , y) ,t (x , y))=14

( 1+ s ) ( 1+ t )

ϕ4 ( x , y )=N 4 ( s (x , y ) , t (x , y ))=14

( 1−s ) ( 1+ t )

with the following changes of variables:s ( x , y )=4 x−2.6t ( x , y )=4 y−1

On Figure 2.2.3-1, one represented in dotted projection on the ellipse of each node. It is observed thatthe Iso-zero of the level set interpolated with the functions of form of the quadrangle is rather far awayfrom the initial ellipse.




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Figure 2.2.3-1 : Error of discretization of the level set

With a quadratic interpolation, on an element quadrangle QUAD8, the approximation of the level set iswritten:

lsnh ( x , y )=−0.4434712ϕ1 ( x , y )−0.1ϕ2 ( x , y )+ 0.2222696ϕ3 ( x , y )+ 0.0408060ϕ4 ( x , y )−0.3304038ϕ5 ( x , y )+ 0.0228051ϕ6 ( x , y )+ 0.1112398ϕ7 ( x , y )−0.2027748ϕ8 ( x , y )

where ϕi i=1,8 are the functions of form associated with the nodes with the quadrangle. Thesefunctions are expressed using N i , classical functions of form on the quadrangle of reference, and the

changes of variables between the real coordinates x , y and coordinates of reference s ,t .

ϕ1 ( x , y )=N 1 ( s (x , y) , t (x , y ))=14

( 1−s ) ( 1−t ) (−1−s−t )

ϕ2 ( x , y )=N 2 ( s (x , y ) , t( x , y))=14

( 1+ s ) ( 1−t ) (−1+ s−t )

ϕ3 ( x , y )=N 3 ( s ( x , y) ,t (x , y))=14

( 1+ s ) ( 1+ t ) (−1+ s+ t )

ϕ4 ( x , y )=N 4 ( s (x , y ) , t (x , y ))=14

( 1−s ) ( 1+ t ) (−1−s+ t )

ϕ5 ( x , y )=N 4 ( s( x , y) , t( x , y))=12

(1−s2) ( 1−t )

ϕ6 ( x , y )=N 4 ( s (x , y ) , t( x , y))=12

(1+ s ) (1−t 2)

ϕ7 ( x , y )=N 4 ( s (x , y ) , t( x , y))=12

( 1−s2 ) ( 1+ t )

ϕ8 ( x , y )=N 4 ( s( x , y) , t( x , y))=12

( 1−s ) (1−t 2)

with the following changes of variables:s x , y =4 x−2.6t x , y =4 y−1




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One then compares the error of discretization between the linear interpolation and the quadraticinterpolation Figure 2.2.3-2: to equal refinement, the quadratic elements are adapted to describean Iso-zero of curved form.

2.2.4 Readjustment of the level sets

In order to limit the problems of integration when the crack passes “close” from a node, a procedure ofreadjustment of the level set normal is installation. So on an edge of the grid, lsn cancel yourself toomuch “close” of a node end, the value of lsn in this node is put at zero. The criterion used for themoment is 1% length of the edge. This value is that used by the software developed by the team ofNicolas Moës to GeM (see the paragraph [§ 1.3.2.3] of [feeding-bottle20]).Algorithm of readjustment of the level set normal:

•buckle on the meshs•buckle on the edges of the mesh, ends A and B

if lsn (A )≠lsn (B )

d=lsn A

lsn A−lsn B

if ∣d∣≤0 .01 then lsn A =0 end ifif ∣d−1∣≤0 .01 then lsn B =0 end if

end if if quadratic element

if lsn A =0 and lsn B =0 then

d=lsn(M ) if ∣d∣≤0 .0001 then lsnM =0 end if

end ifend if

•fine buckles•fine buckles


Figure 2.2.3-2: Interpolation error the Iso-zero: quadratic vs linear



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In the case of a quadratic interpolation, a complementary readjustment is necessary in order to limitand control the situations of multiple cancellation of the level set along an edge. Indeed, not to multiplythe configurations of cutting of the elements nouveau riches under elements of integration, oneauthorizes only two situations of multiple cancellation along the edge of an enriched element. The firstcorresponds if the level set is worthless throughout the edge (thus worthless in its two nodes ends andits node medium). The second is the situation where the level set is worthless on one of the nodes endand the node medium (see Figure 2.2.4-1) . In all the other situations, the level set should be cancelledonly once along an edge. It will be then easy to detect cancellations of the level set along an edge toform under elements of integration. To be brought back in these situations, a readjustment is carriedout when one is in one of the situations represented on Figure 2.2.4-2


Figure 2.2.4-2: Readjustment complementary to the level set

Figure 2.2.4-1: Multiple cancellation of the level set along an edge



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In the two configurations of Figure 2.2.4-2 where one adjusts the level set of the one of Nœuds endsto zero, it is the end of the edge which has smallest level set in absolute value which is selected to beadjusted. Also, when the adjustment to be carried out is considered to be too important, which revealsan important curve of the level set compared to the size of the element, a message of alarmrecommending to use a finer grid is emitted. The selected criterion is the following: the message ofalarm is transmitted since:

∣lsn àajuster∣>0.01∗ maxnoeuds del ' élément

∣lsn∣

This procedure is applied as well for the level set tangent as for the level set normal.

2.2.5 Concept of true signed distance

If the level sets are calculated by analytical functions, the choice of the functions is not single for thesame geometry of crack. Indeed, any function with positive values on a side and negative values of theother is valid. However, if one then wishes that the level set represents the true signed distance, thechoice is single. This concept is important thereafter when one defines the coordinates in the localbase in the bottom of crack using the values of the level sets. It is thus necessary to take care to givethe exact formula of the distance to the crack (expression who is not easy for geometries of complexcracks).

One of the interests of the method by projection is that it provides a field which is the true signeddistance. One could thus also plan to combine the two methods. The data of a simple analyticalfunction would initially make it possible to determine the Iso-zero; then to create a simplified grid of thissurface, which would be used as support with the method by projection. The disadvantage is to createa virtual grid 2D crack which east disjoins grid real 3D. If not, one could also consider a phase of orthogonalisation which transforms an unspecified level setinto a true distance [feeding-bottle21]. Note:If the form of the level-set represented, is not regular enough (for example, has a sharp angle or ajunction), the definition of the distance by orthogonal projection on the Iso-zero, perhaps ambiguousFigure 2.2.5-1.

In the vicinity of the singularity, projection on the surface of the level-set is not possible: the gradientthe Iso-zero of the level-set is discontinuous, therefore the normal with the Iso-zero of the level-setdoes not exist. One indicates then by “remote region”, the area where the calculation of distance byorthogonal projection on the Iso-zero, is to be proscribed.

In the “remote region”, one calculates the distance while returning to his fundamental definition: thedistance of a point M with the Iso-zero is the minimal length connecting the point M at a point ofthe Iso-zero. That is to say C the point corresponding to the top of the right angle on the Iso-zero. For any point M in the “remote region” Figure 2.2.5-1, C check minimal length, i.e.:

d (M , iso_zero)=min(∥MN∥ , N ∈{iso_zero})=∥MC∥ .




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In addition, let us note that the interpolation of level-set regularizes the singularity. As the level-setnonregular is interpolated by polynomial functions §2.2.3, the level-set discretized becomes a regularfunction within the element.

2.2.6 Multi-cracking

The level sets are defined for each crack by the operator DEFI_FISS_XFEM. The list of the cracks isnecessary for the creation of finite elements X-FEM (operator MODI_MODELE_XFEM). During thisphase, “concaténés” fields are created. For example, one created a field of level total set normal to allthe cracks. For each node (fields with the nodes) or each mesh (fields by element), one will seekinformation associated with the crack nearest.

The fields with the “concaténés” nodes created are:•Level set normal,•Level set tangent,•Statute of the nodes (see §3.2.5.1 ), •Base local at the bottom of crack (see §2.3 ).

•Those dregs with under-DECoupage (see § 3.3 ), The fields by “concaténés” elements created are:

•Those dregs with the structures of data for the contact (see R5.03.54 document).

Concerning the propagation [R7.02.13], one can define several cracks on the model and one can givethe list of the cracks which are propagated: with each call of the operator PROPA_FISS, all the cracksgiven is propagated.

Restrictions:The cracks must be suffisamment spaced one of the other (3 meshs without cracks must at leastseparate them). A fortiori, the cracks should not cross either. If not, the introduction of specialenrichments is necessary [feeding-bottle22].

2.3 Base local at the bottom of crackWarning : 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 2018 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)

Figure 2.2.5-1 : difficulty of the definition of a distance byorthogonal projection in the event of singularity of the Iso-zero



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The gradients of the level sets can be used to define the local base in the bottom of crack [feeding-bottle17] [feeding-bottle16]. The gradients are given thanks to the derivative of the functions of forms and the nodal values of thelevel sets.

{∇ lsn j

elt=∑

i

i , j lsni

∇ lst jelt=∑

i

i , j lst i

j=1,3

where them i , j are the derivative of the functions of form compared to the direction j .One thus determines a field of gradients by element. The values are calculated with the nodes of theelements, for each element independently of the others; then to calculate the nodal value one averageon the values obtained by elements with the nodes.

The local base at the bottom of crack {e1 , e2 , e3} is calculated then in any point thanks to the fields ofnodal gradients:

e1=∑i

ϕi∇ lst i

∥∑i

ϕi∇ lst i∥, e2=

∑i

ϕi∇ lsn i

∥∑i

ϕi∇ lsn i∥, e3=e 1×e2

where ∇ lsni and ∇ lst i are the nodal values of the gradients.

Figure 2.3-1 : Base local at the bottom of crack

In 2D one will only have:

e1=

∑i

i∇ lst i

∥∑i

i∇ lst i∥, e2=

∑i

i∇ lsni

∥∑i

i∇ lsni∥

2.4 Determination of the bottom of crack

The bottom of crack is defined by the intersection of the Iso-zero of both level sets. For the calculationof the stress intensity factors (Stress Intensity Factors in English), it is practical to define pointsbelonging to the bottom of crack, which will be used as a basis for the interpolation of SIFs (see theparagraph [§5.1.3]).The selected points are the intersections of the faces of the elements with the curve lsn=0∩lst=0 .These points will be then ordered so as to define a curvilinear X-coordinate along the bottom of crack.

2.4.1 Research of the points of the bottom of crack

The research of the points of the bottom of crack is done in the manner following. One restricts oneselfwith the elements where the level set normal changes sign and where all the nodes are statutes “Ace-Tip” (this concept is defined in the paragraph [§3.2.5.1]).




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That is to say a face of such an element, if this face is a quadrangle, one is brought back to twotriangles.That is to say the triangle ABC

Figure 2.4.1-1 : Face of an element intersected by the bottom of crack

The point is sought M solution of the following system:

{lsnM =0lst M =0

The point is written M in the reference mark (A , AB , AC )M=A1

AB2AC

The system is written then

{(1−ξ1−ξ2 ) lsn(A)+ξ1 lsn(B)+ξ2 lsn (C )=0

(1−ξ1−ξ2 ) lst (A)+ξ1 lst (B)+ξ2 lst (C )=0

⇔{1

2}=[ lsnB−lsnA lsnC −lsn A

lst B−lsnA lst C −lsnA ]−1

{−lsnA−lst A }The point M is retained provided it belongs to the triangle ABC .

In 2D, one uses the same process but directly on the meshs and not on the faces of the meshs (alwaysof the triangles or many quadrangles cut out in triangles). The research of the points of the bottom of crack is accompanied by a strategy of elimination of thepoints M redundant. Indeed, if the face is not a face of the geometrical edge of the structure, it is thencommon to two meshs of the list of the meshs of the bottom, and a point of the bottom can thus bedetected (at least) twice. To check if the new point M detected does not belong already to the list ofthe points P bottom, one carries out a characterization of the point M according to whether it islocated on a node top of the face, an edge of the face or inside the face. One carries out then achecking according to the cases, via a loop on the points P of the same statute than M :

- if M is located on a node top of number NM and that NM=N P ,




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- if M is located on an edge characterized by the numbers of tops N 1M and N 2

M and that

{N 1M+N 2

M=N 1

P+N 2

P

N 1M .N 2

M=N 1P . N 2

P ,

- if M is located inside a face characterized by NNF tops and that

{∑i=1

NNFN i

M=∑i=1

NNFN i

P

∏i=1

NNFN iM=∏i=1

NNFN iP

∑i , j∈(1,NNF) , i≠ jN iM .N j

M=∑i , j∈(1,NNF ) , i≠ jN i

P . N jP

,

then the point M is not preserved. In the contrary case, he is added to the list of the points Pbottom. To free itself from the digital errors which could generate that a point M normally located on anode top sees itself characterized like an interior point on the surface, a procedure “made to vertex” isassociated with the preceding strategy. Thus, if M is detected like a point located on an edge orinside a face, the following algorithm is used:- calculation of the distance from each node top from vis-a-vis the point M ,- each one of these values is placed in the vector Dist who is then sorted smaller value with largest,- if Dist(1)<10−4 .Dist (2) then M is replaced on the node top. If not, it preserves its precedingstatute.An equivalent procedure is installation to replace (if need be) an interior point with the face on an edge. Notice :

Even when one restricts oneself with the elements where the level set normal changes sign andwhere all the nodes are of type “Ace-Tip”, one cannot be sure that the system is invertible. Onecould already limit oneself to the triangular faces where the level sets change sign in the broadsense (0 included), but that would not eliminate all the cases from null determinant. Indeed,when the trace of the bottom of crack on the triangular face is a curve, the system admits aninfinity of solution. This case is not detectable a priori, and only a test on the not-nullity of thedeterminant makes it possible to be freed from such cases. Thus, so on a face the determinant isnull numerically, one does not determine a point of the bottom of crack on this face. If there aresome (inevitably an infinity), the two points solution on the edges of the face are then determinedby another face of the element in question about which the determinant is not-no one.

2.4.2 Orientation of the bottom of crack

The orientation of the bottom of crack is necessary only in 3D.

2.4.2.1 Explanation of the method of orientation of the bottom of crack

At the time of the research of the points of the bottom of crack by the method mentioned in theparagraph [§2.4.1], the points are not inevitably found in an order allowing the immediate definition ofan ordered way and a curvilinear X-coordinate. However, the definition of a bottom of crack orderedand a curvilinear X-coordinate along the bottom of crack is essential to the calculation of G by themethod G-theta.

The method of orientation used is based on the fact that two consecutive points of the bottom belonginevitably to the same mesh 3D.

To explain this method, one will base oneself on the example of a bottom of crack defined by a linecrossing a parallelepipedic structure. One illustrated on the following figure the sight of the top of thegrid of this structure in 3D:




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Figure 2.4.2.1-1 : Sight of top of a grid ofa structure parallelepipedic

The classification of the meshs is arbitrary.

One can see on the figure 2.4.2.1-1 bottom of crack in red made up of 4 points.The whole of the meshs of this structure have their level set normal which changes sign. The 14 meshsrepresented known as are connected to the bottom of crack and have each one at least one of the fourpoints of the bottom.The ordered list of the points of this bottom is 1-2-3-4.

To search the points of the bottom, one makes a loop on the 14 meshs connected to the bottom. Thisloop being made in the order ascending of the indices of the meshs, one finds initially item 1 then items3 and 4 (or 4 then 3) and finally item 2. The following list of the points is thus obtained: 1-4-3-2. Itremains to order this list.

Stage 1 : Research of the points of the bottom by mesh

The principle of the orientation is based on the fact that a mesh connected to the bottom inevitablycontains 1 or 2 points of the bottom.If for example the bottom passes by only one top of a mesh, the latter will contain only one point of thebottom. While returning in a mesh 3D by a face, the bottom must inevitably arise by another point ofthis same mesh. This one contains two points of the bottom then.

Notice :If one or more meshs have more than two points of the bottom, the procedure fails and thebottom is not directed. An element hexahedron HEXA8 can contain for example three points ofthe bottom if this one cut clearly two opposite faces and shaves another face.If the orientation of the bottom is necessary for the continuation of calculations, it is necessary torefine the grid to have only meshs containing to the maximum two points of the bottom.

The first stage of this orientation consists in associating for each mesh connected to the bottom, acouple of indices of points of the bottom their pertaining.

In the example, mesh 1 has only item 1, one associates the couple to him 1,0 . Mesh 2 has items 3and 4, the couple is associated to him 4,3 , etc.

That is to say NMAFON the number of meshs connected to the bottom.

A list is created LISTPT of size 2×NMAFON , containing the indices of the points of the bottom foreach one of NMAFON meshs connected to the bottom.

In our example, the list LISTPT contains 14 couples of indices and could describe itself as follows:

Index of the mesh 1 2 3 4 5 6 7 8 …

Couples associated withthe mesh

1,0 4,3 3,0 3,4 2,0 2,0 2,1 3,2 …

Remarks :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 2018 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)

4

9

101312

11

14 5

1 2 3

12

3

46

7 8



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– Couples 4,3 and 3,4 are equivalent.– The couples containing one 0 will not be considered. Indeed, a point will always belong has atleast a couple of nonworthless indices. For example item 1 belongs to the couple 2,1 .– One detects the points ends while looking to how much couples of nonworthless indices thepoints belong. Item 3 is in the two couples different of nonworthless indices 3,4 and 2,3 .Thus, it is known automatically that item 3 is located between item 2 and item 4. Item 1 fact onlypart of the couple of nonworthless indices 2,1 . Item 1 is thus obligatorily an end of the bottom.

Stage 2 : Research of the ends of the bottom

As one has just mentioned it, the ends of the funds of crack are locatable because only one couple ofindices all nonworthless in LISTPT their index contains. Thanks to this indication, the list is createdPTEXTR including the whole of the points ends.

In the example, item 1 belongs only to the couple 2,1 and the 4 qu point ' with the couple 4,3 .They are then the two ends of our bottom.

Stage 3 : Scheduling of the points of the bottom

This stage corresponds to the orientation of the bottom strictly speaking.To begin scheduling one starts from a point end. This one is the first point of the list PTEXTR . If thebottom is closed (bottom without end), the first point is that having index 1.

In our example, the first point is item 1. To know the following point, it is enough to seek the couplehaving one 1 and one second nonworthless value. In our case, there exists only one couple with the 1with knowing it 2,1 , therefore the second point of the bottom is the 2. Then, item 2 is only in thecouple 2,3 , therefore the following point is the 3. One continues this approach until all the points areordered.One then obtained the list of the items 1-2-3-4.

Notice :If the first found end had been item 4, one would have ordered the bottom in the other direction.One would have obtained the ordered list of the items 4-3-2-1.

2.4.2.2 Case of a multiple bottom of crack

In 3D, the bottom of crack is a line either closed (crack not emerging), or open (emerging crack). Inmost case, the bottom of crack is a continuous line, like that of the circular crack represented onFigure 2.4.2.2-1.




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Figure 2.4.2.2-1 : Case of a continuous bottom of crack

However, it can happen that the bottom of crack is in fact made up of several discontinuous pieces. Itis the case for example circular crack represented on Figure 2.4.2.2-2. In this case, one alwaysspeaks bottom of crack, as the whole of the pieces of the bottom. It is said that the bottom of crack is amultiple bottom. On the example of Figure 2.4.2.2-2, the bottom of crack is composed of the curvedlines BC , DE , FG and HA .

Figure 2.4.2.2-2 : Case of a multiple bottom of crack

Figure 2.4.2.2-3 illustrate the discretization of a multiple bottom of crack (including two pieces). Thefirst piece is composed of items 1 to 5 and the second piece is composed of items 6 to 10.

Figure 2.4.2.2-3 : Multiple Fund of crack

In the case of a multiple bottom, the first two stages of scheduling of the points are unchangedcompared to the case of a continuous bottom.At the time of the third stage, the cracks having a multiple bottom are detected when a second end isreached, without to have ordered the whole of the points of the bottom. One then searches thedeparture of the bottom following while taking a point of the list PTEXTR .

In the case illustrated on the figure 2.4.2.2-3, PTEXTR contains items 1,5,6 and 10. One starts byordering the points from 1 to 5. Then, as it is known that there remain five points to be ordered, onetakes item 6 of PTEXTR to continue the orientation of the bottom.

Notice :In 2D, each point of the bottom of crack represents with him only a bottom.

2.4.2.3 Algorithm used for the orientation of the bottom

That is to say:– NFON the number of points of the bottom,




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– NMAFON the number of meshs connected to the bottom,– LISTPT the list of the couples of indices of the points of the bottom associated with the meshs.

One initializes to 0 a vector of entireties TABPT of length NFON , which will contain the indices ofthe ordered points.

The couple of indices associated with a mesh with index IMA is LISTPT 2×IMA , LISTPT 2×IMA1 .

# the first point of the list of the ordered points:TABPT 1=PTEXTR1

# once the point end used, one puts it at 0, to avoid taking it into account in the event of search foranother end if one has a multiple bottom:PTEXTR 1=0 Buckle on the points of the bottom: IPT=1 with NFON−1 :

◦Buckle on the meshs connected to the bottom: IMA=1 with NMAFON :

One seeks the couple of nonworthless indices where the point of the bottom lately sunken is inTABPT . If it is the case, the second index of the couple is inevitably the point of the followingbottom:

▪If the second index of the couple associated with IMA 0 are worth, one is unaware of thiscouple: to pass to the following mesh IMA1 ▪If the first index of the couple corresponds to that required:

•If IPT2 :◦If the couple of IMA was already found (case of a couple in double):▪to put the second value of the couple at 0 to be unaware of this couple with the nextiteration▪to go to the following mesh IMA1

◦If one deals with a new couple, the second index is the point of the following bottom: ▪ TABPT IPT1=LISTPT 2× IMA1 ▪ to put the second value of the couple at 0 to be unaware of this couple with the nextiteration

▪If it is the second index of the couple which corresponds to that required:•If IPT2 :

◦If the couple of IMA was already found (case of a couple in double):▪to put the second value of the couple at 0 to be unaware of this couple with the nextiteration▪to go to the following mesh IMA1 ◦If one deals with a new couple, the first index is the point of the following bottom:▪ TABPT IPT1=LISTPT 2× IMA ▪ to put the second value of the couple at 0 to be unaware of this couple with the nextiteration

◦If one did not find points to be associated with IPT , it is thus a point end (case of the multiplefunds)




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▪it is checked that IPT is well a point of the list PTEXTR then one puts it at 0 to beunaware of it with the next iteration▪one searches a new point end in the list PTEXTR to begin the new bottom from crack andone passes to the following mesh IMA1




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3 Problem of cracking with X-FEM3.1 Problem general

In this part, one points out the equations of the problem general of a fissured structure. A crack isconsidered c in a field ∈ℜ

3 delimited by ∂ of external normal next . The lips of the crack arenoted

1 and 2 external normals n1 and n2 . The displacement and stress fields are respectively

noted and u .A quasi-static loading is imposed on the structure via a density of voluminal forces f , of a density ofsurface forces t on Γ t and of a density of surface forces g on the lips. The solid is embedded onΓ u .

Figure 3.1-1 : Notations of the problem general

The strong form of the equilibrium equations and the boundary conditions is written:

∇⋅= f dans ⋅next=t sur t

⋅n1=g sur

1

⋅n2=g sur 2

u=0 sur u

éq 3.1-1

We place ourselves within the framework of small deformations and small displacements, for which therelation deformation-displacements is written:

= u=∇ su éq 3.1-2

where ∇ s is the symmetrical part of the gradient and the tensor of the deformations.

A linear elastic material is considered1. The law of behavior of the solid is written:

=C : dans éq 3.1-3

where C is the tensor of Hooke.

The principle of virtual work is written:

1 For a non-linear material, nothing changes in the formulation presented here. Only the calculation of theconstraints according to the deformations and the internal variables changes, which is transparent in thisdocument.




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∫σ u : ε v d=∫

f⋅vd∫Γtt⋅vdΓ∫Γ c

g⋅vdΓ ∀ v∈H 01 éq 3.1-4

where H 01 is the space of Sobolev of the functions whose derivative is of square integrable, being

cancelled on ∂Ω .

3.2 Enrichment of the approximation of displacement

The principal idea is to enrich the base by the functions of interpolation thanks to the partition of theunit [feeding-bottle23]. The classical approximation finite elements is pointed out:

uh x = ∑i∈N n x

a ii x

where them a i are the degrees of freedom of displacement to node I and i functions of formassociated with node I. N n x is the whole of the nodes whose support contains the point x . Onecompares the support of a node I to the support of the functions of form associated with this node, i.e.

with the whole of points x such as i x ≠0 .

The enriched approximation is written:

uh(x )= ∑i∈N n (x)

a iϕi(x )+ ∑j∈N n (x)∩K

b jϕ j( x)H j ( lsn(x ))+ ∑k∈N n (x)∩L

∑α=1

4

ckαϕk (x )F

α(lsn( x) , lst (x))

This expression is made up of 3 terms. The 1er term is the continuous classical term. 2ème and 3ème

terms are terms enriched. Being in the middle of method X-FEM, these terms are clarified in thefollowing paragraphs.

3.2.1 EnrichesmeNT with a function of selection of field (2ème term)

Let us suppose that the interface Γ c partitionne the field such as Ω=Ω+∪Ω- . If Γ c is a crack,one partitionne same manner the field Ω while extending virtually Γ c .

In order to represent the jump of displacement through Γ c , one introduces the function of selection offield or function characteristic of field H j ( x ) [feeding-bottle76] defined by:

Si x j∈Ω+ , H j(x )={ 0 si x∈Ω+

−2 si x∉Ω+

Si x j∈Ω- , H j( x)={ 0 si x∈Ω-

+2 si x∉Ω-

While making use of the level set normal, quantity H j (lsn ( x ) ) is worth 0 if the point x and thenode x j are same with dimensions crack and ±2 if not. The coefficient “2” is introduced to have asimpler writing of the average jump of displacement along the interface.

b j are the degrees of freedom nouveau riches. K is the whole of the nodes whose support isentirely cut by the crack (nodes represented by a round on Figure 3.2.1-1).

Notice : by abuse language, one will call also the functions of selection of field, functions “Heaviside”.But it will be necessary to refer to the definition above, to represent the approximation of thediscontinuity of the field of displacement.




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Figure 3.2.1-1 : On the left, the “round” nodes are enriched by the “square” Heaviside functionand nodes by the singular functions (topological enrichment). On the right, the “square” nodes are

enriched by the singular functions (geometrical enrichment).

3.2.2 Enrichment with a function of selection of field for a connection of cracks

Let us suppose a connection of two discontinuities, which partitionne space in three fields such asΩ=Ω1∪Ω2∪Ω3 . If it is about a connection of two cracks, one partitionne in the same way the field

Ω by virtually extending the three branches of cracks.

Figure 3.2.2-1 partitioning of the field by a junction of crack

For a reducing single Y branch, the generalization of the functions of selection of fields leads to thefollowing writing:

uh(x )= ∑i∈N n (x)

a iϕi(x )+ ∑j∈K∩Ω1

b j ,1ϕ j(x )χΩ2+b j ,2ϕ j(x )χΩ3

+ ∑j∈K∩Ω2

b j ,1ϕ j( x)χΩ1+b j ,2 ϕ j(x )χΩ3

+ ∑j∈K∩Ω3

b j ,1ϕ j(x )χΩ1+b j ,2ϕ j(x )χΩ2

Where χΩiare function of selection of fields defined by:


Ω1

Ω2

Ω3



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χΩi(x )={ 2 si x∈Ωi

0 si x∉Ωi

However, the topological description of three related fields (Ω1, Ω2 and Ω3) is not commonplace, takinginto account the only furnished information by the level-sets. The levet-sets make it possible torepresent a scalar change of sign through a discontinuity. Whereas we wish to represent thepartitioning of space in the vicinity of discontinuity.

From an “elementary” point of view, we note that the field of sign of level-sets read in vectorial form,adhesive roughly with the partitioning which we want to carry out. This partitioning, thanks to thevectorization of the scalar level-sets, then makes it possible advantageously to re-use the structures ofdata of Code_Aster. However, only elementary partitioning is insufficient, to represent the “ddl field”(χΩϕI).

Note:In fact, the “ddl field” corresponds to the collection of the partitions of field Ω on the whole of thesupport of node I. On the support of each node, it is thus essential of concaténer the elementarypartitions corresponding to a “ddl field” given. This operation is not commonplace since the fields ofsign evolve from one element to another (as the scalar fields of signs are not prolonged on all theelements, to locate on some elements of interest the definition of a crack, also called “band close” toenrichment). An algorithm dedicated to the concatenation of elementary partitioning, was developed tosolve this fundamental difficulty.

Figure 3.2.2-2 : partitioning of an element multi-fissured using the field of“vectorized” sign. For each field, the first scalar component of the vector signs,corresponds to the characterization of the discontinuity of the crack n°1 and thesecond scalar component of the vector signs, corresponds to the discontinuity

of the crack n°2.

Construction of the functions of sign or junctions:

Let us point out the construction of the functions of Heaviside sign or functions of type junction[feeding-bottle70]. It is about a Heaviside function which “is truncated” on the level of the connection.The nodes nouveau riches are represented on Figure 3.2.2-3.




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-1

+1

0

Figure 3.2.2-3 : Enrichment for the second crack. The round nodes are enriched by the functionjunction, which is worth +1, -1.0.

The value of this function depends on the levels set normals of the 2 cracks. One considers the sign ofthe level set normal of crack 1, on the side where crack 2 is defined signlsn1 fiss2 (in practice,

one looks at the sign of a point pertaining to the field of lsn1 in which crack 2 is, cf operand JUNCTIONDoc. [U4.82.08]). There is then the function of enrichment junction for the crack 2 which is written:

J 2x ={H lsn2 x si signlsn1 fiss2H lsn1x ≥00 sinon

It is possible to connect a third crack on the first (for example modelling an intersection):

J 3x ={H lsn3x si signlsn1 fiss3H lsn1x ≥00 sinon

If one wishes to connect the third crack on the second, it is necessary to take account of the field ofdefinition of the first, one will thus have:

J 3x ={H lsn3x si ∀ i∈[1,2] , signlsni fiss3H lsni x ≥00 sinon

If one generalizes the approach with a crack N who connects on the cracks of a unit K , the whole ofthe cracks then is defined P , which contains at the same time all the cracks of K and all the crackson which possibly the cracks connect of K . The function junction is written then in a general way:

J N x={H lsnN x si ∀ i∈P , signlsni fissnH lsni x ≥00 sinon

An example of configuration is presented Figure 3.2.2-4. On this example one builds a tree of

connectivity of the cracks. One deduces from this tree that for N=3 , one has K=2 and P=[1,2] .In the same way for N=5 , one deduces K=[3,4 ] and P=[1,2 ,3,4 ] . Thus one hasJ 5x=H lsn5x on the hatched field from the figure, and zero elsewhere.




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1

2 3 4

5

1

24

3

5

Figure 3.2.2-4 : Network of cracks on the left, tree of hierarchy on the right, the shaded zonecorresponds to the field where the function of enrichment of crack 5 is not worthless.

Exploitation of the functions of sign for the assembly of the ddls fields:

To build the ddls fields defined in the preceding paragraph, we propose to re-use the information of the field ofHeaviside signs to define an elementary partitioning. The algorithm being too much complex to explain in termsof structures of data, we give a graphic translation of the algorithm. Here for example fields of Heaviside signsconcaténés which one wishes to exploit to build elementary partitioning:

Simple crack Simple junction Multiple junction

The case of a mono-crack being rather easy, we will be delayed in detail on case of a simple junction. In thefollowing graphic explanation, we propose to change prospect compared to elementary description suggestedabove. The algorithm will be described from the point of view of the support of the node, which is adapted moreto describe the nodal functions of enrichment.

From the point of view of the node, information on the fields of signs is richer than the idyllic case describesabove. Indeed, the node sees information on the close cracks, i.e., located in the first and the second band ofelements contiguous to the support of the node. In the support of the node, there thus exists a statute of cracks(cf [D4.10.02]) to discriminate the cracks intersecting the support of the node and the cracks close notintersecting the support to the node.

This difficulty makes noncommonplace association between the fields of signs and the partitions of fields. Thisassociation is necessarily local and must take into account the pollution of the fields of sign related to the closebands quoted previously. On the example set in the table below, it is necessary to associate the fields with thefield of sign in the following way:

• the field is associated Ω1 ↔ with the field of sign {+1 ,0 ,*} • the field is associated Ω2 ↔ with the field of sign {-1 , -1 ,*} • the field is associated Ω3 ↔ with the field of sign {-1 ,+1 ,*}




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Partitioning in field Field of associated sign

For to compress the information of the vectors signs, one prefers to transform the vectors signs usingthe function of coding reversible following:

code (P )= ∑ifiss=1

nfiss

3nfiss−ifiss (He (P , ifiss)+1 )

where,• ifiss is the number in the vector field of sign,

• P indicate the point running (a node or a point of gauss),

• nfiss is the length of the field of sign.

Figure 3.2.2-5 : the vectors signs have variable lengths, which leads todifferent identifiers for the same field. These identifiers then constitute a

second level of under-partitioning.

Taking into account the variability the length nfiss vectors signs, a field can have a code different froman element with another, like magazine Figure 3.2.2-5. For example, the field Ω1 have two identifiers

(5 and 6). These identifiers can be seen like two under-partitions field Ω1 .




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This under-partitioning is not awkward as long as information on this under-partitioning is gathered onthe level of the node, for the assembly of the “ddls” field. To identify these two ddls fields in eachelement, it is enough then to concaténer information on the under-partitions:

• If a under-partition of the complementary field is found, information is stored in the dedicatedsite (structure of data per node and element),

• If not, there is no under-partition for this element.

The result of such an elementary loop is summarized in the table below. In each element one searchesa under-partition of each field complementary to the blue node. If no under-partition is found in theelement, one marks in the site dedicated a cross ‘X’ (or -1 for example).

For the node in blue pertaining to the field Ω1 , the evaluation of the characteristic functions χΩ2 or

χΩ3 is obvious above taking into account information on the partitioning of the table. In each element,

the first component (of the concaténé field) informs about the identifier of the first ddl field χΩ2, the

second component (of the concaténé field) informs about the identifier of the second ddl field χΩ3 :

• If the identifier of the field to which the point of Gauss belongs corresponds to the componentk∈[1,2] stored with Nœud, then the characteristic function associated with the componentk takes the value “+2”. Let us specify again that the component k=1 is associated with the

ddl field χΩ2 and the component k=2 is associated with the ddl field χΩ3

. It is important

to stress that this position does not change an element with another, to ensure the adjacencyof partitioning. For example, that is to say the collection of the identifiers {0.1} corresponding tothe field Ω2 . It is exactly the information which is stored in first position in the structure of data

to the “blue” node, in the elements intersecting the field Ω2 .

• If not, the characteristic function takes zero value if the identifier at the point of gauss does notcorrespond to the component to node in the element considered.

Note:

1. One can notice that the ddl field χΩ1 is not to treat in the example above. Indeed, to

represent the two discontinuities introduced by the connection of two cracks it is necessary toenrich with two discontinuous functions them nodes whose support is intersected by doublediscontinuity. One then chooses to enrich with the ddls associated with the fieldscomplementary to the field to which belongs it node. This choice is well adapted, taking intoaccount the problems of conditioning detailed with the §3.2.6.

2. In Code_Aster, one prefers to store the information of the Heaviside sign and that of theidentifier of field at the point of Gauss in the subelements of integration.

3.2.3 Enrichment with the singular functions (3ème term)




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In order to represent the singularity in bottom of crack, one enriches the approximation with functionsbased on the asymptotic developments by the field of displacement in linear elastic breaking process[feeding-bottle25]. These expressions were given for a plane crack in infinite medium.

u1=12 r2 K1cos

2

−cos K2sin2

κ2cos u2=

12 r2 K 1 sin

2 κ−cosK 2 cos

2 κ−2cos

u3=12 r2 K 3sin

2

=E21

et =3−4 en déformations planes .

éq. 3.2.3-1

The assumption of plane constraints cannot be retained because no fissured plate is in situation ofplane constraints in the vicinity of the singularity, when one places oneself at a finite distance from theskin of the hull.

The base making it possible to describe these fields comprises 4 functions:

{ r cosθ2, r cos

θ2

cosθ , r sinθ2, r sin

θ2

cosθ } .Like:

{cosθ2

cosθ=−sinθ sinθ2+cos

θ2

sinθ2

cosθ=sinθ cosθ2−sin

θ2

the following base then is chosen2 :

F={ r sinθ2, r cos

θ2, rsin

θ2

sinθ , r cosθ2

sin θ} .where r ,θ are the polar coordinates in the local base at the bottom of crack (see Figure 2.3-1 andFigure 3.2.3-1).These coordinates can be expressed easily thanks to the level sets, since:

r= lsn ²lst ² , θ=arctan lsnlst , θ∈[−π2 , π2 ] In practice, the computing function rather is used atan2 lsn ,lst who returns the principal value of the argument of the complex number lst , lsn expressed in radians in the interval [−π ,π ] .

For points being located exactly on the lower lip lsn=0 , the function atan2 0, lst give a beingworth angle π . In theory, atan2 −0,lst allows to obtain −π as expected, but it is not numericallyalways the case. For stage this disadvantage, one rather uses the following expression for the angleθ :

θ=H lsn ∣atan 2 lsn , lst ∣, θ∈ [−π ,π ] where H lsn is the value of the Heaviside function. Thus, when one is on the lower lip, the value−π is well reached.

2 For non-linear laws of behavior, this choice is preserved, although this base does not make it possible to findthe exact solution of the problem.




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Figure 3.2.3-1 : Polar coordinates in the local base

It is noted that only the 1era function of the base is discontinuous through the crack. The other functionsare added only to improve the precision. These functions are the solutions of Westergaard, analyticalasymptotic solutions of an elastic problem of rupture in 2D. This base is well adapted to the cases 3D[feeding-bottle16] [feeding-bottle18], at least for the cracks whose bottom is rather regular. Thesefunctions are known as “singular” because their derivative are singular in r=0 .

Figure 3.2.3-2 : Functions of enrichment in bottom of crack




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Figure 3.2.3-3 : Derived from the functions of enrichment

ck are the degrees of freedom nouveau riches. L is the whole of the nodes whose support is

partially cut by the bottom of crack (nodes represented by a square on Figure 3.2.1-1). That means thatonly one lays down elements is enriched around the bottom by crack. This enrichment is called“topological”.

3.2.4 Geometrical enrichment

As of the first papers on X-FEM [feeding-bottle24], it is notified that a geometrical criterion rmax canbe defined to determine the nodes nouveau riches by the singular functions (see Figure 3.2.1-1):

L={noeuds tels que rrmax } The first studies of convergence were carried out in 2000 within the framework of the GFEM [feeding-bottle26], with taking into account of several layers of elements nouveau riches in bottom of crack.

When convergence is studied, one is interested in the evolution of the error compared to the degree ofrefinement of the grid. Generally, one indicates by h a length characteristic of the elements of thegrid, and one seeks to determine the parameter α called rate (or speed or order) of convergence,such as the relative error erel form is written:

erel=∥u−uh∥H 1 ≈h0Chα

where C is a constant independent of h .Since

log erel≈ logCα log h the parameter α seems the slope of the right-hand side log erel according to log h when h tendstowards 0. Stazi et al. [feeding-bottle27] studies the convergence of the error in energy for an infinite plate with aright crack, in mode I, for linear and quadratic formulations. He notices that the quadratic one improvesthe error, but not the rate of convergence. Béchet et al. [feeding-bottle28] this observation confirms andshows that a fixed zone of enrichment makes it possible to find an almost optimal rate of convergence.

In parallel, Laborde et al. [feeding-bottle29] the question deepens, and tests the rates of convergencefor the polynomial formulations of a higher nature. Moreover, he makes improvements in order to findan optimal rate, even supra a convergence. Table 3.2.4-1 gather the results got by Laborde for variousalternatives of X-FEM, and this for polynomial approximations of degree k=1,2 ,3 .

FEM X-FEM X-FEM (F. A.) X-FEM (D.G.) X-FEM (tokenentry)

P1 0.5 0.5 0.9 0.5 1.1P2 0.5 0.5 1.8 1.5 2.2P3 0.5 0.5 2.6 2.6 3.3

Table 3.2.4-1 : Order of convergence of the various alternatives of X-FEM

The first column corresponds to the orders of convergence of the classical finite element method for aproblem of cracking. Taking into account the singularity, the speed of convergence is in h whateverthe degree k . The simulations 2D carried out on a test problem: rectilinear crack on a square in modeI of opening shows that X-FEM does not improve the rate of convergence. This can be explained bythe fact why topological enrichment relates to only one lay down elements in bottom of crack. The zoneof influence of this enrichment is thus strongly related to h . Thus, when h tends towards 0, the sizeof the zone of influence of enrichment also tends towards 0. The idea which seems natural is thenmore not to limit the zone of enrichment to only one lay down elements, but to extend it to a zone offixed size, independent of the refinement of the grid. 3ème column of Table 3.2.4-1 present the resultsgot with this method known as X-FEM F. A. (for Fixed enrichment Area). One finds almost the hoped




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rates of convergence ( α=k for an approximation Pk ). However, conditioning is degraded comparedto usual method X-FEM. In order to find an acceptable conditioning, Laborde proposes to gather thedegrees of freedom nouveau riches by the singular functions. In light, instead of having ddls differentnouveau riches for each enriched node, they are globalized in order to have of them only one bysingular function and end of crack (but in 3D, one does not see well how it goes). With thisarrangement, conditioning is largely improved, but the rates of convergence are weaker (X-FEM D.G.).The problem comes from the transition metals between the enriched zone and zones it not-enriched.According to Laborde, the phenomenon is explained by the effect of the partition of the unit whichcannot be used on these elements partially nouveau riches. To mitigate this defect, an ultimate versionis proposed: X-FEM token entry (for Pointwise Matching). The displacement of the nodes on the borderbetween the enriched zone and zones it not-enriched are imposed equal. Thanks to this stickingtogether, hoped rates of convergence are obtained (even a light super-convergence).

Notice :

For the approximation polynomial of degree 2,3k , Laborde and al. [feeding-bottle29] asmuch of others uses functions of form of degree k for the classical terms and nouveau richesby the functions of discontinuity (2nd term), so that the jump is of degree k . On the other hand,for the terms nouveau riches by the singular functions (3rd term), it is enough to use the linearfunctions of form to collect the singularity in bottom of crack and step to deteriorate theconditioning of the matrices too much.

Currently, in Code_Aster, only an approximation of degree 1 is possible, and the fact of informing or nota value of ray for the zone of enrichment (keyword RAYON_ENRI) allows to place itself respectivelywithin framework X-FEM (F. A.) or classical X-FEM.

3.2.5 Enrichment in Code_Aster3.2.5.1 Enrichment (statute) of the nodes

To know if a node is enriched by the Heaviside function (node of the type “Heaviside) or by the singularfunctions (node of type “ace-tip”), one calculates the min and the max of the level set normal on all thenodes belonging to the support of the node (the concept of “support” is defined in the paragraph[§3.2]), and one calculates the min and max of the level set tangent on all the points belonging to thesupport of the node considered where the level set normal is cancelled.

j∈K⇔( minx∈Nn( j)

(lsn( x )) maxx∈Nn ( j )

( lsn( x ))<0) et ( maxx∈N n( j )∩lsn(x )=0

( lst ( x ))<0)k∈L⇔( min

x∈Nn( j)( lsn( x )) max

x∈Nn( j )( lsn(x ))≤0) et ( min

x∈Nn( j )∩lsn( x)=0

( lst ( x )) maxx∈N

n( j )∩lsn(x )=0

( lst ( x ))≤0)Similar ideas appear in [feeding-bottle17], but it would seem that certain cases were not taken intoaccount. These expressions are the result of the first efforts [feeding-bottle30] which aimed atdetermining the types of enrichment only using the level sets. The reader will find there the figurescorresponding explanatory.

In the case of a geometrical enrichment in bottom of crack, the selection criteria of the nodes is thefollowing:

k∈L⇔ lsn x ²lst x ²≤rmax

Concerning the choice of the value of the ray of enrichment, nothing is clearly indicated in the literature.It would seem however that a being worth ray enters 1/5 and 1/10 length of the crack is a relevantchoice.

Recent studies showed that geometrical enrichment strongly degrading the conditioning of the matrix ofrigidity, it had to be limited in a zone restricted around the bottom of crack, in waiting of treatmentmaking it possible to improve conditioning. One proposes an alternative, which is halfway betweentopological and geometrical enrichment: an enrichment on n layers [feeding-bottle31]. In this case,




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one is calculated rmax (single for each crack) according to the user datum amongst layers, then thepreceding formula is applied. Algorithm of choice of the enrichment of the nodes:

That is to say MAFIS the whole of the meshs on which the level set normal is cancelled

•buckle on all the nodes P grid

initialization of the max and min of the level sets

•buckle on the meshs of MAFIS containing the node P

•buckle on the edges of the mesh

are A and B two ends of the segment

if lsn ( A )=0 then

actualization so necessary of maxlst and minlst with lst A end ifif lsn B =0 then

actualization so necessary of maxlst and minlst with lst B end ifif lsn A lsn B 0 then

• in the case of a linear interpolation

C=A−lsn( A )

lsn(B )−lsn( A )(B−A ) éq. 3.2.5.1-1

• in the case of a quadratic interpolationOne determines the coordinate within the space of reference ξpoint of intersection C with the curve lsn=0 on the edge. Itslst is then given by:

lst (C)=lst (A )∗ξ∗(ξ−1)

2+lst (B)∗(ξ+1)∗ξ

2−lst (M)∗(ξ−1)∗(ξ+1)

éq. 3.2.5.1-2

actualization so necessary of maxlst and minlst with lst C .end if

•fine buckles

•buckle on the nodes tops of the mesh

actualization so necessary of maxlsn and minlsn

•fine buckles




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•fine buckles

if minlsn .maxlsn0 and maxlst≤0 then P∈K

topological case of enrichmentif minlsn .maxlsn≤0 and minlst .maxlst≤0 then P∈L

geometrical case of enrichment:

if lsn P 2 lst P 2≤rmax then P∈L

•fine buckles

To obtain the equation [éq. 3.2.5.1 - 1], as well as the value of the level set tangent at the point C , thecurvilinear X-coordinate as a preliminary is determined s such as

C=As B−A thanks to the fact that the level set normal is cancelled in C , that is to say

lsn C =lsn A s lsn B −lsn A =0

One from of deduced the expression from the point C as well as the value of the level set tangent inC :

lst (C )=lst ( A )+s (lst (B )−lst ( A ) )

=lst ( A )−lsn (A )

lsn (B )−lsn ( A )(lst (B )−lst (A ) )

To obtain, in the quadratic case, the equation [éq. 3.2.5.1 - 2], as well as the value of the level settangent at the point C , one determines the coordinate within the space of reference as a preliminaryξ point Clsn is interpolated quadratically along the edge. It is thus a question of solving a polynomial of thesecond degree:

0=lsn(A)∗ξ∗(ξ−1)

2+lsn(B)∗(ξ+1)∗ξ

2−lsn(M )∗(ξ−1)∗(ξ+1)

The equation lsn=0 have a single solution on the edge because lsn A lsn B 0

Once obtained the coordinate ξ point C , they are interpolated lst points A , B and M to obtainlst point C :

lst (C)=lst (A )∗ξ∗(ξ−1)

2+lst (B)∗(ξ+1)∗ξ

2−lst (M)∗(ξ−1)∗(ξ+1)

Notice :

The same node can belong to the units K and L .

3.2.5.2 Enrichment (statute) of the meshs

In Code_Aster, it is necessary to define types of precise finite elements, and not to multiply the numberof the possibilities, the choice was made to define 3 types of finite elements X-FEM: the elements“Heaviside”, elements “ace-tip” and the elements mixed “Heaviside and ace-tip”. If the mesh has at least a node of the type “Heaviside”, then it is a mesh “Heaviside”.If the mesh has at least a node of type “ace-tip”, then it is a mesh “ace-tip”.If the mesh has at least a node “Heaviside” and at least a node “ace-tip”, or if the mesh contains atleast a node “Heaviside and ace-tip”, then it is a mesh “Heaviside and ace-tip”.




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Let us note GRMAEN1 the meshs “Heaviside”, GRMAEN2 meshs “ace-tip” and GRMAEN3 themeshs “Heaviside and ace-tip”.That is to say the mesh i , and Ni the whole of the nodes j mesh i .si (∃ j∈Ni , tel que j∈K ) alors i∈GRMAEN 1si (∃ j∈Ni , tel que j∈L ) alors i∈GRMAEN 2

si (∃( j , k )∈Ni2 , tels que j∈K et k∈L ) ou (∃ j∈Ni , tel que j∈K∩L ) alors i∈GRMAEN 3

It is noticed that all the nodes of an element will be affected same characteristics and sameenrichment, but it is not inevitably what is desired. It is thus necessary to cancel the degrees offreedom nouveau riches “in excess”.

3.2.5.3 Cancellation of the degrees of freedom nouveau riches “in excess”

One saw in the preceding paragraph which a mesh of the type “Heaviside” can comprise for exampleone node of the type “Heaviside”, other nodes of the mesh being classical nodes not requiring anyenrichment. However these nodes will be affected degrees of freedom of the mesh “Heaviside”,therefore of degrees of freedom nouveau riches. Consequently, it is necessary to wrongly carry out acancellation of these degrees of freedom nouveau riches. Cancellation allows in the facts continuouslyof passing from a zone enriched to a not enriched zone and makes it possible to make cohabit twostandard elements (within the meaning of they divide a common border), one enriched, the other notenriched. The not enriched variable is the same one on the common border and the degrees offreedom corresponding to enrichment are put at zero for the element which is enriched on this sameborder (but not elsewhere in this same element). This way of making avoids having to solve thequestion of the “blending elements” whose one can find a treatment in [feeding-bottle73]. Several cases arise:•degrees of Heaviside freedom to be cancelled with the classical nodes of a mesh Heaviside or mixed•degrees of freedom ace-tip to be cancelled with the classical nodes of a mesh ace-tip or mixed•degrees of Heaviside freedom to be cancelled with the nodes ace-tip of a mixed mesh•degrees of freedom ace-tip to be cancelled with the Heaviside nodes of a mixed mesh.

The technique of cancellation of these degrees of freedom is explained in detail in [R5.03.54, §4.4].

3.2.6 Conditioning related to enrichment3.2.6.1 General information on conditioning XFEM

Conditioning, noted 10δ corresponds to the relationship between largest and the smallest eigenvalue

of a system to be reversed. For a calculation in double precision with a digital error in 10−15 , the

relative error obtained on calculation is about 10−15 . One must thus check the condition δ9 toguarantee a digital precision about 10−6 .

Geometrical enrichment strongly degrades the conditioning of the matrix of rigidity [feeding-bottle29],[feeding-bottle28]. Béchet and al. [feeding-bottle28] propose a technique of orthogonalisation of thedegrees of freedom during the calculation of the elementary matrices of rigidity in order to improveconditioning of the assembled matrix. Laborde and al. [feeding-bottle29] explain that bad conditioningis due to the fact that the selected base of enrichment does not train a free family locally. They thuspropose to put only one degree of freedom for these functions on all the zone of enrichment and toconnect displacements to the limit between zones enriched and not enriched in order to find optimalrates of convergence. The problem of conditioning is besides such as with quadratic elements itbecomes impossible to get results, without setting up one of the techniques [feeding-bottle29],[feeding-bottle28]. Indeed, for these elements, bad conditioning is due not only to the singular part ofenrichment, but also to Heaviside enrichment, when the crack passes very near to a node. Onepresents the evolution amongst conditioning as the interface approaches the nodes of the grid on thefigure 3.2.6.1-1, for linear and quadratic elements respectively. The values of this figure are veryapproximate. On the one hand one does not take into account the elements of the vicinity. In addition




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these values are obtained in a coarse way. In the case of the cube cut in a corner for example (on theright on the figure), one considers the rigidity of the small cube of the corner on side 10− instead ofthe small tetrahedron of the corner. Finally the conditioning of a total problem is generally larger thanlocal conditioning and increases when the grid is refined. By taking into account all theseconsiderations, one in practice obtains conditionings of an order 2 to 4 times superiors with those of thefigure 3.2.6.1-1.

43 33

3 2 4 4

56 679

Figure 3.2.6.1-1 : the distance to the node top nearest standardized by the length on the beingworth side 10− one shows the dependence of number of conditioning, compared to the

parameter for the linear partly higher and quadratic elements partly lower.

3.2.6.2 Estimated criteria to improve conditioning

Technique of readjustment of the level set evoked in the paragraph [§2.2.4] allows to secure this badconditioning. While noting 10−γ the distance from the point of intersection of the level set with nodetop nearest standardized by the length on the side, the readjustment with 1% length of edge made inthe paragraph [§2.2.4] corresponds to γ=2 . This readjustment must act sufficiently quickly so thatconditioning is not deteriorated too much, but not for values of γ too much weak in order not to disturbthe system by moving in a nonrealistic way surface of cracking. For quadratic elements hexahedrons, ifit is necessary that 10−1592 that is to say about 10−5 , one obtains 92=10 that is to say areadjustment with 13% length of edge. It is thus not possible in this case to reasonably activate therethe readjustment of the level sets at the top, so that conditioning is not deteriorated.

Under these conditions, one sets up a criterion allowing to detect if a degree of Heaviside freedomposes problem. If the criterion is checked, the degree of freedom is eliminated by zero setting, as in theparagraph [§ 3.2.5.3 ].

The principle is the same one as the voluminal criterion suggested by Daux [feeding-bottle22] for thejunctions. The idea is to look for each node whose support is cut by the level set, the report of the sizesof the zones of share and others of the level set on this support (affected zones of a value of Heavisidebeing worth ±1). If:

min V−1 ,V1 V support

≤10− éq, 3.2.6.2-1

the degrees of Heaviside freedom of the node concerned are put at zero in all the directions.




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Figure -3.2.6.2-1 : illustration of the installation of the voluminal criterion. The support of thenode cut by the level set is illustrated. One in the case of compares gray and white volumescompared to the total volume of the support of the node (meeting of gray and white volumes) acrack (left) and of a junction (right).

If this criterion is relevant for the linear elements (triangles, tetrahedrons) with a value of from 4, it isnot satisfactory for the multilinear elements (quadrangles, pyramids, pentahedrons, hexahedrons) andquadratic. Indeed the value of from 4 conduit to be eliminated from the degrees of freedom whichwould not owe the being, which disturbs the solution, whereas higher values of degradeconditioning. In order to take into account these elements, one uses a criterion of rigidity, which isbased on a comparison of rigidities of the support and either simply of volumes. For each node whosesupport is cut by the level set, one looks for this support the report of rigidities of the zones of shareand others of the level set (affected zones of a value of Heaviside being worth ±1). So in a node nwhere se are the subelements of its support, one a:

min∑se−1∫

se

∥n ,X∥2 dse ,∑se1

∫ se

∥n ,X∥2dse

∑se∫

se

∥n ,X∥2d se

≤10−

éq 3.2.6.2-2

where n ,X is the derivative of the function of form to the node n in the total direction X , thendegrees of Heaviside freedom of the node n are put at zero in all the directions. One will notice thatthe behavior is not present in the criterion of rigidity of éq 4.3-2, the criterion having been standardized.That allows saycriminer in advance degrees of freedom to be eliminated, without knowledge of theproblem to be solved (non-linearities, plasticity, contact, etc). This criterion, very near to a criterion ofconditioning, leads us to choose values of ranging between 8 and 10. In practice, we take a valueof from 9.

The criterion describes in éq 4.3-2, quantifies in order of magnitude, the compromise between quality of the solution and conditioning. As this criterion is valid only in order of magnitude, it is not inevitably relevant, to calculate all the integrals exactly. In the programming Aster, one thus approximates the integral calculus on the subelements ([feeding-bottle72]):

∫ se

∥n , X∥2dse≈ ∥n ,X G

se∥∞

2 V se éq 3.2.6.2-3

where G se indicate the barycentre of the subelement, V se the volume of the subelement.

Instead of calculating the integral on the subelement, one evaluates the derivative of functions of formin a point. This strategy makes it possible to collect in order of magnitude the criterion of elimination oféq 4.3-2, with low costs of calculation.

This approximation proves to be robust and inexpensive with linear elements.

On the other hand, with quadratic elements, the derivative of the functions of form admit roots.Calculation in a point is risky: the barycentre perhaps a root of the derivative of functions of form,where the estimate will be worthless, but not the integral on the subelement. This bad estimate cangenerate eliminations by chance, which disturbs the solution and is not acceptable.

One thus reinforces the estimate for the quadratic elements, by adding other points of evaluation,besides the barycentre:

∫ se

∥n , X∥2d se≈ maxPi∈se

∥n ,X P i∥∞2 V se

éq 3.2.6.2-4

So that the evaluation is relevant, the points are sufficiently well distributed: one considers theequidistant points between Nœuds tops and the barycentre (for example, 3 points for a tri6).




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{P i }={G se}∪{NodeiG

se

2} éq 3.2.6.2-5

However criticisms following, can be made compared to these estimated criteria:

• on the one hand, they do not guarantee the control of conditioning right before elimination. As

long as elimination is not carried out, conditioning increases in 10n×γ (figure 3.2.6.1-1), whichcan lead to the stop of the direct solvor, in an arbitrary way. This phenomenon is exacerbatedwith the quadratic elements, even figure 3.2.6.1-1.

• in addition, an analysis of these various criteria is made in [feeding-bottle72]. It shows thateliminations lead to an absence of convergence of the error in energy on a simple example ofhomogeneous compression of a cube crossed by a tilted interface. The suggested solutionconsists in replacing the elimination of the degrees of Heaviside freedom by aorthogonalisation of the local matrices of rigidity, idea which comes from pre-conditioner X-FEM of [feeding-bottle28]. In order to overcome this last difficulty it is necessary to considermatrices of rigidity local at least in triples precision [feeding-bottle of it72].

To address, the first criticism, i.e. to dam up the exponential increase in conditioning, we set up inCode_Aster, the pre-conditioner proposed by Béchet and al. [feeding-bottle28].

3.2.6.3 Pre-conditioner XFEM for the matrices

It is about an automatic and algebraic procedure, for “orthogonaliser” the degrees of freedomassociated with a node XFEM. Indeed, the functions of enrichment XFEM are based on the nodalfunctions Φi to describe, that is to say the discontinuity of displacement by the functions jump HΦi ,

that is to say the bottom of crack by the singular functions FαΦi . The functions introduced byenrichment XFEM are not orthogonal with the functions of form in each node XFEM. Moreover, thefunctions of enrichment XFEM and the functions nodal, share the same support: it arrives of thesituations where functions XFEM approach the nodal functions, at the point to become almostcolinéaires.

Information on the colinearity is transported in the matrix of rigidity K : the conditioning of the matrix ofrigidity increases if the colinearity increases at least in a node XFEM. From a more formal point of view,the matrix of rigidity (cf.§3.5.1) drift of the discretization of a positive symmetrical bilinear form.Therefore, there exists a product-scalar ⟨. , .⟩K such as, the term of the matrix of rigidity K i , x

representing the colinearity enters the nodal function Φi and the function of enrichment F xΦi ,rewrites itself:

K i , x=⟨Φi , F xΦi ⟩K éq 3.2.6.3-1

Béchet [feeding-bottle28] the construction of a pre-conditioner proposes such as:

K→ K=P cT K P c éq 3.2.6.3-2

One transforms the original matric system then, in the following way:

K u= f →{K u= ff =P c

T fu=Pc u

éq 3.2.6.3-3

Thus, Béchet [feeding-bottle28] built the new matrix K to observe the condition of followingorthogonality:

K i , x=⟨Φ i , F xΦi ⟩ K=⟨PcΦi , Pc F xΦi ⟩K=0 éq 3.2.6.3-4




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In other words, the new functional base (P cΦi , Pc F xΦi) (by the basic change Pc ) is orthogonal,

on the support of each node XFEM. Consequently, the new matric system to reverse K u=P cT f is

conditioned better independently of the position of the interface, cf §3.2.6.2.

The construction of K is not single. [feeding-bottle28] proposes to use the factorization of Cholesky ofthe local matrices of rigidity, associated with each node XFEM:

K loci = [

⟨Φi ,Φi⟩K ⟨Φi , HΦi ⟩K ⟨Φi , FαΦi⟩K

⟨Φi , HΦi⟩K ⟨HΦi , H Φi⟩K ⟨HΦi , FαΦi⟩K

⟨Φi , FαΦi⟩K ⟨HΦi , F

αΦi⟩K ⟨FαΦi , FαΦi⟩K

] éq 3.2.6.3-5

The matrix K loci is symmetrical definite positive. She thus admits one factorized of Cholesky, i.e.,

which there exists a higher triangular matrix S i such as: K loci =S i

T S i éq 3.2.6.3-6

In a contiguous configuration of storage optimal of the degrees of freedom, [feeding-bottle 28 ]chooses then a diagonal pre-conditioner per block, such as:

Pc = [Pc

1 0 0 ... ... ...

0 Pc2 0 ... 0 ...

0 0 Pc3 ... ... ...

... ... ... ⋱ ... ...

... 0 ... ... P ci ...

... ... ... ... ... ⋱] éq 3.2.6.3-7

where Pci= {S i

−1 si i est un noeud XFEMI d sinon

éq 3.2.6.3-8

with S i−1 the reverse of the factorized matrix of Cholesky. The new matrix of rigidity K check

with the local scale of the support of node XFEM:

K loci=(Pc

i )TK loci Pc

i=(S i

−1 )TK loci S i

−1=(S i

−1 )T(S i

T S i )S i−1= I d éq 3.2.6.3-9

I d indicate the matrix identity. I d thus observe the condition of orthogonality of the equation3.2.6.3-4 .

In practice, it is necessary to put on the scale the new matrix K loci with the rest of the matrix of rigidity.

One thus prefer the following pre-conditioner with the pre-conditioner 3.2.6.3-8 :

Pci = √ scal×S i

−1 éq 3.2.6.3-10

where scal is a coefficient of scaling such as: scal=max(∣diag (K )∣)+min (∣diag (K )∣)

2

Consequently, the new packaged local matrix observing the condition of orthogonality 3.2.6.3-4 , is:




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K loci

= (P ci )TK loci P c

i= scal×I d éq 3.2.6.3-11

In addition, the use of a factorization of Cholesky is risky if the conditioning of the matrix of local rigidity

K loci degrades itself. In the event of failure of the factorization of Cholesky, one orthogonalise the local

matrix using a SVD (decomposition in singular values):

K loci = U i DiU i

T éq 3.2.6.3-12

where U i is an orthogonal matrix and Di is a diagonal matrix with strictly positive values.

One carries out then alternatively, the choice of pre-conditioner, following:

Pci = √ scal×U i√Di

−1 éq 3.2.6.3-13

It is easy to check that this choice makes it possible to observe the condition of orthogonality 3.2.6.3-4 :

K loci =(Pc

i )TK loci Pc

i=(√scal×U i√Di−1)

T(U i DiU i

T ) (√ scal×U i √Di−1 )= scal×I d

In short, the construction of the pre-conditioner of Béchet [feeding-bottle 28 ] proceeds in 4 stages:

1.the extraction of the local matrices K loci in the matrix of rigidity (i.e. matrices blocks

associated with the ddls carried by the nodes XFEM n° i ).

2.the calculation of the local matrices of prepacking Pci (cf equation 3.2.6.3-10 or 3.2.6.3-13 ),

3.assembly of the pre-conditioner (cf equation 3.2.6.3-7 ),

4.finally, the transformation of the matric system (cf equations 3.2.6.3-2 and 3.2.6.3-3 ).

Note:

Let us note that the transformation of the matric system presents a notable data-processing difficulty inCode_hasster, taking into account the complexity of the structure of data of storage of the matrices (cf [D4.06.10 ] and [ D4.06.07 ] ).

3.3 Under-cutting

A special attention must be carried during digital integration of terms of rigidity and second member ofan element crossed by the crack. Indeed, on an element crossed by a crack, the gradients ofdisplacements can be discontinuous, and in this case the digital integration of Gauss-Legendre on thetotality of the element is not applicable. In order to replace itself under classical conditions of regularity,it is advisable to carry out an integration on fields where the intégrande is at least continuous. For anelement crossed by a crack, it is thus necessary to integrate separately on both sides of the crack (thisappears for the 1era time in [feeding-bottle24] for the 2D and in [feeding-bottle32] for the 3D). Severalprocedures possible, and are easily put in work in 2D. The difficulties appear with the 3D.

One seeks under-to cut out under-tetrahedrons an unspecified voluminal element (tetrahedron,pentahedron, hexahedron) cut by a surface. It is pointed out that this cutting is used only with ends asintegration, it is purely virtual and no node is added to the grid. The grid is not in nothing modified.

For the pentahedrons and the hexahedrons, the number of possibilities being too important and the toocomplicated configurations, one prefers to be reduced to cutting tetrahedrons. One “thus condenses” alarge number of possibilities of cutting to some configurations. In other words, we formulate the“clustering” of a set of big size (many configurations of intersection) towards a set of small (reducednumber of configurations of cutting in a tetrahedron).




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In fact, the algorithm of “condensation” built a bijective mapping, polyhedral relative, towards someconfigurations of cutting of reference. This application perhaps broken up into two bijective operations:

1 – a bijection (clarifies) towards the cutting of a tetrahedron: One réalisE thus a preliminary phasewhich consists in cutting out in a systematic way them quadrangles, pyramids, pentahedrons andhexahedrons in tetrahedrons. This bijection is not single. As one can see it on Figure 3.3.1-1, thereexist two bijections to divide a quadrangle into two triangles (in the same way for a pyramid), 6bijections to divide a pentahedron into 3 tetrahedrons and 6 bijections to divide a hexahedron into 2pentahedrons which are then divided into tetrahedrons. In the case of the hexahedron, one thus twiceobtains each configuration of cutting. The maximum number of distinct bijections to divide a

hexahedron into 6 tetrahedrons is thus of 63

2=108 .

Figure 3.3-1: Division of one quadrangle in two triangles (on the left), division of a pentahedronin 3 tetrahedrons (in the medium) and division of a hexahedron in 2 pentahedrons (on the right)

In § 3.3.1 with § 3.3.3 , one presents the bijection selected by default for the division of thenonsimpliciaux elements in tetrahedrons (or triangles for the case 2D). It is about a bijection, because one builds an application (invertible) which associates the relativeclassification of the nodes tops in a under-tetrahedron, with the absolute classification of the nodes inthe grid. The direct application is stored in the table of connectivity (even [D4.10.02] : structures ofdata XFEM).2 – a bijection (implicit) towards a configuration of cutting of reference: on the one hand, one identifiesthe configuration of cutting of reference corresponding to the tetrahedron to be cut out, on the otherhand, one turns over “geometrically” the tetrahedron, “to superimpose it” on this configuration of cuttingof reference (for example Figure 3.3.1-2). In light, one identifies the nodes of the under-tetrahedron tothe nodes of the element of cutting of reference (it is the direct application of the bijection). One“then condenses” the multiple possibilities of cutting, related to the various possibilities of scheduling ofthe edges and the points of intersection. Thereafter, this operation facilitates the scheduling of the nodes mediums of the quadratic under-tetrahedrons for data-processing storage. Indeed, scheduling new nodes mediums does not dependany more of the faces or the edges where they are calculated, but of only one configuration of cuttingof reference. Consequently, the new nodes calculated mediums are necessarily scheduled samemanner, in the under-tetrahedron of cutting of reference. Then, these nodes mediums are positionedon the edges of the tetrahedron by the opposite application (of the identification of the nodes above).The bijection used is thus tacit. The procedures of scheduling and calculation of the points mediums are clarified Ci below.

3.3.1 Preliminary phase of cutting of the hexahedrons to bring back itself totetrahedrons




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Figure 3.3.1-1 : Division of a hexaèdre (hexa8) in tetrahedrons

A hexahedron is divided then into 6 tetrahedrons, indexed in the following table:

hexahedron tetrahedron

N1 N2N3 N4 N5 N6 N7 N8

N7 N4 N3N1 N1 N6 N2 N3 N3 N6 N7 N1 N6 N1N5 N7 N4 N7 N8 N5 N4 N5 N1 N7

Table 3.3.1-1 : Division of a hexahedron in tetrahedrons

It is noted that this cutting is the choice carried out by default but ON could have chosen anothermanner of cutting out a hexahedron in 2 pentahedrons like another way of cutting out a pentahedron in3 tetrahedrons. If the element is quadratic (HEXA20, PENTA15, …), one uses the same subdivision in tetrahedrons, asthat clarified above. In addition, it is necessary to take account of the nodes mediums on the new edges generated bycutting: these nodes mediums coincide with the nodes mediums of the complete element. One thusadds virtually with the element, the nodes mediums necessary to find the complete element of Figure3.3.1-2.




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Figure 3.3.1-2: Division of a quadratic hexahedron (hexa 20) via the complete element (hexa 27)

3.3.2 Preliminary phase of cutting of the pentahedrons to bring back itself totetrahedrons

Figure 3.3.2-1 : Diagram of division of a pentahedron in tetrahedrons

A pentahedron is divided then into 3 tetrahedrons, indexed in the following table:




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pentahedron tetrahedron

N 1N 2N 3N 4N 5N 6 N 5N 4N 6N 1 N 1N 2N 3N 6 N 6N 2N 5N 1

Table 3.3.2-1 : Division of a pentahedron in tetrahedrons

For a quadratic pentahedron (penta15) one preserves the subdivision above. In addition, it is necessary to take account of the nodes mediums on the new edges generated bycutting: these nodes mediums coincide with the nodes mediums of the complete element. One thusadds with the element, the nodes mediums necessary to find L‘complete element Figure 3.3.2-2.

3.3.3 Preliminary phase of cutting of the pyramids to bring back itself to tetrahedrons


Figure 3.3.2-2: Division of a quadratic pentahedron



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Figure 3.3.3-1 : Diagram of division of a pyramid in tetrahedrons

A pyramid is divided then into 2 tetrahedrons, indexed in the following table:

pyramid tetrahedron

N 1N 2N 3N 4N 5 N 1N 3N 4N 5 N 1N 2N 3N 5

Tableau 3.3.3-1 : Division of a pyramid in tetrahedrons

For a quadratic pyramid (pyra13) one preserves the subdivision above. In addition, it is necessary to take account of the nodes mediums on the new edges generated bycutting: these nodes mediums coincide with the nodes mediums of the complete element. One thusadds with the element, the nodes mediums necessary to find the complete element Figure 3.3.2-2.

Notice :

Lsubdivision of a quadrangle has EN two triangles is similar to the subdivision of a pyramid intwo tetrahedrons.

3.3.4 Under-cutting of a tetrahedron under-tetrahedrons

The tetrahedron of reference is defined on Figure 3.3.4-1. One determines the points of intersectionPi between surface lsnh=0 and edges of the tetrahedron.

That is to say n the number of points of intersection Pi . At each point of intersection Pi , two entireties are associated: Ai and NSi

• Ai is the number of the edge on which is Pi (for example if Pi is on the edge of endsN2−N3 , then Ai=4 ). If Pi coincide with a node top of the tetrahedron, Have is worth 0,


Figure 3.3.3-2: Division of a quadratic pyramid



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• NSi is the number of the node top Dyears the case where Pi coincide with a node top of thetetrahedron (for example if Pi coincide with N3 , then NSi=3 ). If Pi is on an edge, NS0 are worth.

Notice :

The product of Ai by NSi 0 are always worth.

The points of intersection are then sorted according to the order ascending of Ai . The points ofintersection coinciding with nodes top will thus be found at the beginning of the list.

Figure 3.3.4-1 : Tetrahedron of reference

The approximation of the level set using the functions of form of the tetrahedron, surface lsnh=0 isthen a plan with linear elements, and a possibly curved surface of geometry, with quadratic elements. The problem is thus reduced to the cutting of a tetrahedron by a surface. Let us examine the variouspossible cases according to the value of n (many points of intersection Pi ). One canalready eliminate the commonplace cases where no under-cutting is necessary:

•when n3 the trace surface in the tetrahedron is a top or an edge. If surface lsnh=0 is not aplan, the geometry of the level-set is then approximated by the edge of the tetrahedron.

•when n=3 and that the 3 points of intersection are points tops, the trace surface in thetetrahedron is a face of the tetrahedron. If surface lsnh=0 is not a plan, the geometry of thelevel-set is then approximated by the face of the tetrahedron.

In these two cases, one obtains only one under-tétra, who corresponds to the tetrahedron.

Figure 3.3.4-2 : Case without under-cutting

3.3.4.1 Three points of intersection of which two points tops

P1 and P2 are inevitably the two points top (thus A1=A2=0 ). According to A3 , number of edgecorresponding to P3 , one can determine the 2 ends of this edge, are E1 and E2 .




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Figure 3.3.4.1-1 : Case general where n=3 including 2 points top

One obtains a under-tetrahedron of each with dimensions interface, that is to say with final the 2 under-tetrahedrons.

2 under-tetrahedronsP1 P2P3 E1 + P1P2P3 E2

Table 3.3.4.1-1 : Under-tetrahedrons

To reduce the possibilities of cutting, (since there exist at least 6 possibilities of cut taking into accountthe 6 edges in the under-tetrahedron) one builds a bijection towards one single element of cuttingFigure 3.3.4.1-2. Nodes A , B , C , D , E , F , G , H the unknown factors of scheduling of the cuttingof the under-tetrahedron define. The unknown factors of scheduling will be associated in a single waywith the nodes of the element relative. This association builds a bijection (implicit).Let us recall that,- scheduling is important from a data-processing point of view, to ensure the storage of the virtual meshof under-cutting. In particular, to locate the point mediums and points of intersection in the structures ofdata XFEM,- the nodes of number ranging between 1000 and 1999 represent the points of intersection, thesenodes are calculated with the procedure described with the §4.2.3 of [D4.10.02].- the nodes of number higher than 2000 represent the new nodes mediums resulting from cutting,these nodes are calculated with the procedure described with the §4.2.3 of [D4.10.02]

Notice :

For this configuration of particular cutting, the construction of the bijection is not entirelydeterministic. Indeed, the 2 under-tetrahedrons are symmetrical compared to the plan of cutting.From a topological point of view, the unknown factors { A , D , E } is symmetrical with theunknown factors { B , F , G }. To make deterministic the cutting of a grid given, we chosefollowing arbitrary convention: to take A like the first node of the edge of the first point ofintersection.By analogy to the molecular configurations, cutting potentially will generate two configurationsknown as “enantiomers”.


Figure 3.3.4.1-2: element of cutting forconfiguration 4 points of intersection

including 2 Nœuds tops



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3.3.4.2 Three points of intersection of which a point top

P1 is inevitably the point top (thus A1=0 and A2≠0 ). Edges A2 and A3 have a joint node,E1 , and 2 different nodes E2 and E3 .

Figure 3.3.4.2-1 : Case general where n=3 of which a point top

One obtains a under-tetrahedron of with dimensions, and other a pyramid which one divides into 2under-tetrahedrons, is with final the 3 under-tetrahedrons.

3 under-tetrahedronsP1 P2P3 E1 + P1 P2P3 E3 + P1 P2E2 E3

Table 3.3.4.2-1 : Under-tetrahedrons

To reduce the possibilities of cutting, one builds a bijection towards one single element of cuttingFigure 3.3.4.2-2. Nodes A , B , C , E , F , G , H the unknown factors of scheduling of the cutting ofthe under-tetrahedron define. The unknown factors of scheduling will be associated in a single waywith the nodes of the element relative. This association builds a bijection (implicit).

3.3.4.3 Five points of intersection including two points tops

This case of cutting corresponds to a shaving configuration. It is a typical case of the preceding cuttingfor which surface lsn=0 master key in more by another node top. Cutting is then strictly identical topreceding cutting (Paragraph [§3.3.4.2]). This configuration of cutting is represented Figure 3.3.4.3-1


Figure 3.3.4.2-2: element of cutting for configuration 3 pointsof intersection including 1 node top



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with on the right the configuration at 3 points of intersection and a point top to which one brings backoneself.

3.3.4.4 Three points of intersection of which no point top

At least 4 possibilities of cut, indexed are distinguished Figure 3.3.4.4-1. These various possibilities arenot treated on a case-by-case basis.

One builds a bijection towards one single element of cutting Figure 3.3.4.2-2. Nodes A , B , C , D ,E , F , G the unknown factors of scheduling of the cutting of the under-tetrahedron define. Theunknown factors of scheduling will be associated in a single way with the nodes of the element relative.This association builds a bijection (implicit).

Example of procedure of identification:In each listed configuration Figure 3.3.4.4-1, one associates the unknown factors of scheduling Nœuds{ A , B ,C , D} with Nœuds {N 1,N 2,N 3,N 4} under tetrahedron. Like the node A belongs to the

three cut edges, one easily identifies it with the node corresponding in each listed configuration. Then,one identifies then {B ,C , D } since Nœuds B , C , D belong respectively to the first, second, thirdintersected edge, and are different from the node A previously identified. It results the table from itfrom correspondence Table 3.3.4.4-1This procedure of identification makes it possible to build a bijection implicitly.

Figure 3.3.4.4-1 : Configurations of a tetrahedron with 3 intersected edges


Figure 3.3.4.3-1: Shaving configuration at 5 corresponding healthy configuration and points ofintersection (on the left) (on the right)



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Of Figure 3.3.4.4-2 with Figure 3.3.4.4-1 A , B ,C , D N 1,N 2,N 3,N 4 A , B ,C , D N 2,N 1,N 3,N 4 A , B ,C , D N 3,N 1,N 2,N 4 A , B ,C , D N 4,N 1,N 2,N 3

Table 3.3.4.4-1 : Identification of the nodes of the element of cutting to the nodes of the under-tetrahedron in each configuration

3.3.4.5 Four points of intersection of which a point top

This case of cutting corresponds to a shaving configuration. It is a typical case of the preceding cuttingfor which surface lsn=0 master key in more by one of the nodes tops. Cutting is then strictly identicalto preceding cutting (Paragraph [§3.3.4.4]). This configuration of cutting is represented Figure 3.3.4.5-1 with on the right the configuration at 3 points of intersection to which one brings back oneself.


Figure 3.3.4.4-2: element of cutting for configuration3 points of intersection and no node top




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3.3.4.6 Six points of intersection including two points top

This case of cutting corresponds to a shaving configuration. It is still a typical case of the precedingcutting for which surface lsn=0 master key in more by two of the nodes tops. Cutting is then strictlyidentical to preceding cutting (Paragraph [§3.3.4.4]). This configuration of cutting is represented Figure3.3.4.7-1 with on the right the configuration at 3 points of intersection to which one brings back oneself.

3.3.4.7 Four points of intersection

At least 3 possibilities of cut, indexed are distinguished Figure 3.3.4.7-1. These various possibilities arenot treated on a case-by-case basis.

One builds a bijection towards one single element of cutting Figure 3.3.4.7-2 . Nodes A , B , C , D ,E , F , the unknown factors of scheduling of the cutting of the under-tetrahedron define. The unknownfactors of scheduling will be associated in a single way with the nodes of the element relative. Thisassociation builds a bijection (implicit).

Figure 3.3.4.7-1 : Configurations of a tetrahedron with 4 intersected edges





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3.3.4.8 Five points of intersection of which a point top

This case of cutting corresponds to a shaving configuration. It is a typical case of the preceding cuttingfor which surface lsn=0 master key in more by one of the nodes tops. Cutting is then strictly identicalto preceding cutting (Paragraph [§ 3.3.4.7 ]). This configuration of cutting is represented Figure 3.3.7-1with on the right the configuration at 4 points of intersection to which one brings back oneself.

3.3.4.9 Four points of intersection of which a point top

This configuration of cutting corresponds to a shaving configuration. It is not brought back to anyconfiguration previously described. One distinguishes it from the configuration described in Paragraph[§3.3.4.5]) (which also comprises 4 points of intersection of which a point top) because it node top notintersected on stops shaving belongs only to two edges intersected (instead of 3 in the case describedin the Paragraph [§3.3.4.5]). One distinguishes at least 24 possibilities from cut, of which 3 areindexed in Figure 3.3.4.9-1. These various possibilities are not treated on a case-by-case basis.


Figure 3.3.4.7-2: element of cutting for configuration 4 pointsof intersection and no node top

Figure 3.3.4.8-1: Confishaving guration at 5 corresponding healthy configurationand points of intersection (on the left) (on the right)



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Figure 3.3.4.9-1 : Configurations of a tetrahedron with 4 points of intersection of which a point top

One builds a bijection towards one single element of cutting Figure 3.3.4.9-2 . Nodes A , B , C , D ,E , F , the unknown factors of scheduling of the cutting of the under-tetrahedron define. The unknownfactors of scheduling will be associated in a single way with the nodes of the element relative. Thisassociation builds a bijection (implicit). This configuration of cutting give rise to 5 under-tetrahedronschildren.

3.3.5 Multi-cutting

When one wants to model junctions, intersections or simply that two cracks are enough close to cut outthe same element, it is necessary to be able to divide the element into fields which respect allintroduced discontinuities.

The strategy selected consists in sequentially cutting out the element several times. It is a strategywhich has the virtue to be rather fast to implement, because the cutting of a tetrahedron of reference bya crack generates under-tetrahedrons which can in their turn being regarded as tetrahedrons ofreference for cutting by the following crack.

The problem is that one does not optimize the total number of subelements generated, which is likelyto be very high if one redécoupe more than 3 times in 3D. To solve this problem, it would be necessaryto be able to directly cut out any type of element (hexahedron, pentahedron, pyramid, tetrahedron) incombination of all these type of elements. That requires rather heavy developments and is thus not


Figure 3.3.4.9-2: element of cutting for configuration 4 pointsof intersection of which a node top



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possible. Another strategy would consist in gathering the subelements by zone and finding the numberof points of Gauss (and the weights) optimal per zone. One would not store any more the subelementsbut directly the zones with the points of associated gauss.

3.3.6 Maximum number of subelements

In order to correctly dimension the structures of relative data to under-cutting, it is advisable todetermine the maximum number of subelements generated by the phase of under-cutting, according tothe type of the initial mesh.One considers in this paragraph which the element is cut out only by only one fissures.

3.3.6.1 Case of the tetrahedron

The case generating more a large number of subelements that is described in the paragraph [§3.3.4.7],which leads to 6 subelements.

3.3.6.2 Case of the pentahedron

A pentahedron being divided as a preliminary into three tetrahedrons, one could think that more a largenumber of generated subelements is that where each one of these three tetrahedrons is in its turnunder-cut out in 6 subelements; what a final cutting in 18 subelements would involve. However, such acase is impossible. To the maximum, on the three tetrahedrons, two will be cut out in 6 subelementsand only one will be cut out in 4 subelements, outcome with 16 subelements.


Figure 3.3.5-1: Example of multi-cutting in 2D



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3.3.6.3 Case of the hexahedron

As previously, the case where all the six tetrahedrons each one under-are cut out in 6 subelements isimpossible. The maximum case is that which corresponds to the case evoked preceding paragraph:two pentahedrons deduced from the hexahedron (see Figure 3.3.1-1) under-are cut out each one in 16subelements. The number of subelements is thus (6+6+4)+(6+6+4)=32 .

3.3.6.4 Case of multi-cutting

A finite element XFEM can be cut out by 4 cracks to the maximum in CodeAster. The maximumnumber of under generated elements is then difficult to evaluate and one raising of the maximumnumber of under element would be too high. Indeed, if one takes the maximum number of underelement generated by only one fissures, that is to say 32, and that one considers that the 32tetrahedrons can give rise to 6 tetrahedrons for each following crack, one obtains 32∗6³=6912 . Thisnumber of under elements of integration for a hexahedron is completely unattainable because it goeswithout saying that the tetrahedrons resulting from the 1er cutting all will not be cut out by the followingcracks, but it is difficult to display the best raising. One makes the choice to fix the maximum number ofunder elements for the elements multi-fissured at 4 times the maximum number of under elements foronly one crack is 32∗4=128 in 3D and 4∗6=24 in 2D. These numbers can easily be exceededfor elements cut by 3 or 4 cracks. In this case, the procedure of cutting returns an error and the user isinvited to refine the grid used in order to prevent that finite elements are not striated with cracks, whichwould involve big problems of conditioning in addition (see paragraph [§3.2.6]).

3.3.7 Under-cutting 2D

One uses a method comparable to cutting 3D for cutting 2D. The quadrangles will be subdivided insame triangles them under cut out according to the passage of the crack.

The quadrangles in 2 triangles are cut out:

Figure 3.3.7-1 : Under-cutting of a quadrangle in triangles: linear(on the left), quadratic (on the right)

quadrangles triangles

N1 N2N3 N4 N1 N2N4 N2 N3N4

N1 N2N3 N4 N5 N6 N7 N8 N1 N2N4 N5 N9 N8 N2 N3N4 N6 N7 N9

Then one recuts the triangles according to the passage of the crack. One carries out the same sortingas in 3D for the points of intersection Pi .

Notice :

The node N9 is not a node of the grid. It is added to make connectivity between the quadraticquadrangle and triangles. It is defined as being the medium of the segment N2N4 (and not the




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central node of the quadrangle). The value of the level set normal which is allotted to him isobtained by interpolation of lsn nodes N1 with N8 . A readjustment can then be necessary toensure the unicity of the solution of the equation lsn=0 along the diagonal (cf readjustment ofthe level set [§2.2.4]) at the time of the cutting of the elements children under elements ofintegration. Prior to any cutting, one thus carries out a loop on the edges of the elements children(that was already carried out for the edges of the elements parents during the readjustment ofthe level set) to detect the pathogenic situations and to carry out the readjustments which areessential, as described exactly with [the §2.2.4].

When one has 1 or 2 points of intersection which fall only on the tops, no cutting is necessary.

Figure 3.3.7-2 : Case without under-cutting

If 1 or 2 of the points of intersection does not fall on a top, the triangles are recut.

If there is 1 point top and 1 point not top, P1 is forcing top. One determines the tops of the edge ofthe 2E not N2 and N3 to cut out the triangle in 2 pennies triangles P1 N2P2 and P1 N3P2 . Ifthe triangle is quadratic, one determines the points mediums of the edge of the 2E not Q1 and Q2and that of the crack Q3 .

Figure 3.3.7-3 : Case of linear cutting (on the left) and quadratic (on the right)with 2 points of intersection including 1 top

With 2 points not top, one must cut out in 3 triangles. The triangles are obtained N1P1 P2 ,P1 P2N3 and P1 N2N3 . N1 being the point close to the 2 points of intersection according to thesorting carried out. If the triangle is quadratic, one determines the points mediums of the edge of the 1er

not Q1 and Q2 , those of the 2E not Q3 and Q4 , that of the crack Q5 and that of the new edgeQ6 .




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Figure 3.3.7-4 : Case of linear cutting (on the left) and quadratic (on the right)with 2 points of intersection without top

3.3.8 Algorithms of under-cutting

In the paragraphs §3.3.1 with §3.3.7, we explained the geometrical cutting of the elements under-cells of integration (Tetra10 in 3D, Tri6 in 2D), without explaining how coordinated points ofintersection and new nodes mediums in the virtual grid are calculated. In this paragraph, we will thusdescribe the calculation algorithms of the coordinates of the new points, generated by cutting.

For each type of points, there exists a specific algorithm of positioning:

The positioning of the points of intersection makes it possible to delimit an explicit border,

although the interface/crack is defined by an implicit equation lsnh=0 . Precisely, the place ofthe points of intersection is calculated by taking the intersection between the edges of the gridand the Iso-zero of the level-set. The points of intersection are then stored in a structure ofdata auxiliary, in completion of the points of the grid [D4.10.02].

The positioning of the points mediums makes it possible to improve the description of acurved level-set, in the case of quadratic elements. In the presence of elements on curvedboard, the curve of the edges of the quadratic element is preserved, by positioning nodesmediums rigorously on the edges at the time of cutting. On the other hand, a cutting withelements on right board would deteriorate the curve of the original element. To collect thecurve of the level-set and the curved edges of quadratic elements, one thus calculates severaltypes of points mediums (see § Error: Reference source not found). Several algorithms areimplemented, to collect the various types of points possible mediums. These new pointsmediums are then stored in a structure of data auxiliary, in completion of the points of thegrid [D4.10.02].

Thereafter, the following notation is adopted:

• ξ=(ξi)i∈[1, dim] indicate the coordinates of reference in the element relative. These coordinates

generally vary enters [0,1 ] or enters [−1 ,1 ] . Let us recall that the functions of forms orpolynomials of interpolation, are parameterized in Aster, thanks to the coordinates of reference[R3.01.01],

• X=( X i )i∈[1, dim] indicate the punctual coordinates in real space. If the point running is Nœud,

they are the geometrical coordinates of the node in the grid. If the point running is not a node,it is the geometrical coordinates interpolated in the element relative

X (ξ)= ∑1⩽ j⩽nnop

X jΦ j(ξ) .

3.3.8.1 Calculation of the points of intersection

NR. Moës proposes to pose the problem of research of the points of intersection within the space ofreference relative. In light, to calculate the real coordinates of a point of intersection, one searches the




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coordinates within the space of reference as a preliminary ξ . One deduces then the real coordinatesthanks to the interpolation from the geometrical coordinates in the element relative.

To calculate the point of intersection P between an edge AB and the Iso-zero, one solves thefollowing problem:

P∈ABlsnh (P )=0

Figure 3.3.8.1-1 : Research of the point of intersection I1 on the edge I2 I 3 who is curved

within the space of reference. One takes then in account curve of I2 I 3 with the node medium

M 2

Exception: For the quadratic elements multi-fissured, in the element of reference relative, the edgeAB is not necessarily right. The node medium M edge AB is not necessarily on the segment[A ,B] if the side AB is the side of under element of integration generated by cutting in accordancewith a first crack (see Figure 3.3.8.1-1). It is then necessary to take into account the possible curve ofthe edge; one has thus P∈AB⇔ξP=(2t−1)(t−1)ξA+ t (2t−1)ξB+ 4t(1−t)ξM with t∈[0 ,1] .As a result, the posed problem amounts searching t∈[0,1] such as

lsnh (ξP=(2t−1)(t−1)ξA+ t (2t−1)ξB+ 4t (1− t)ξM )=0 .

For linear or quadratic modelings mono-fissured, the point P necessarily belongs to the edge[A ,B] and thus: ξP=(1−t)ξA+ t ξB with t∈[0 ,1] .As a result, the posed problem, amounts searching t∈[0, 1] such as lsnh (ξP=(1−t)ξA+ t ξB )=0. One then seeks the root of a polynomial equation since the functions of interpolation of the level-setare polynomials. This equation is then solved using the algorithm of Newton-Raphson (below detailed).




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Like f : t→ lsnh(t ) is C2 and strictly monotonous in the vicinity of the point of intersection (if not, the

level-set becomes parallel to the edge AB and does not intersect it), the convergence of thealgorithm of Newton is quadratic, therefore optimal.

Algorithm of Newton-Raphson for the calculation of a point of intersection of the Iso-zero withthe edge AB :

• Reading of the coordinates of reference of the nodes tops A and B and of the possible nodeM medium: by construction, these nodes are nodes of the element child which one cuts outand their coordinates are thus known.

• Initialization of Newton-Raphson:

◦ For linear modelings, one initializes research with the linear approximation along the edge:

t 0=lsn(A)

lsn(A)−lsn(B)

◦ For quadratic modelings, one initializes research with the exact solution of the problemalong the edge of the element child. This solution is given by the solution of a polynomialequation of the second degree. The coefficients of this polynomial of the second degreeare given by the values of lsn with the nodes ends and medium of the edge of underelement child. Since lsn change strictly sign between the ends of this edge, it exists asingle solution with the equation lsn=0 on the edge. The equation to be solved has thefollowing form:

lsn(A )+(4∗lsn(M)−3∗lsn(A)−lsn(B))∗t0+2∗(lsn(A)+lsn(B)−2∗lsn(M ))∗t0 ²=0

Among the two possible solutions of this quadratic equation one retains only that whichsatisfies: t 0∈[0 ,1] . This value constitutes a good initialization and it is even the finalsolution of the problem when the edge of the element child on which one carries outresearch coincides with an edge of the element relative. Indeed, the L lsn is theninterpolated only by the nodes ends and medium of the edge of the element relative. Thisinterpolation being quadratic, initialization provides the exact solution. On the other hand,when the edge of the element newborn does not coincide with an edge of the elementrelative (edge internal Figure 3.3.8.1-2 ) , initialization does not provide the exact solutionof the problem because lsn is interpolated by all the nodes of the element relative alongthe edge of the element child.


Figure 3.3.8.1-2: Intersection of the curve lsn=0 with an edgeof the element child



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• As long as n⩽ itermax to make: ◦

ξn={(1−t n)ξA+t nξB enlinéaire etmono−fissuré(2tn−1)(t n−1)ξA+t n(2tn−1)ξB+4t n(1−tn)ξM enquadratiquemulti−fissuré

◦ Calculation of the function: lsnh(ξn)= ∑

1⩽ j⩽nnop

lsn jΦ j(ξn)

◦ Calculation of the derivative: dlsnh(ξn)/dt= ∑

1⩽ j⩽nnop

lsn j∇Φ j(ξn)⋅v

◦ Checking of the slope: if ∣dlsnh(ξn)/dt∣<10−16 then left with code error.

◦ Calculation of the increment: Δn=lsnh(ξn)

dlsnh(ξn)/dt

◦

◦ Calculation of the new position: t n+1=t n−Δn

◦ Checking of the assumption AB : if t n∉[0,1 ] then left with code error.

◦ Criterion of stop on the increment: if ∣Δn∣<10−8 then fine of the loop.

• Calculation of the coordinates of reference of the point of intersection:

ξ∞={(1−t∞)ξA+ t∞ ξB en linéaire et mono−fissuré(2t∞−1)(t∞−1)ξA+t∞(2t∞−1)ξB+4t∞(1−t∞)ξM enquadratiquemulti−fissuré

• Checking of the assumption P∈ELREFP : if ξ∞∉ELREFP then left with code error.

• Calculation of the real coordinates by interpolation of the geometrical coordinates:

X (ξ∞)= ∑1⩽ j⩽nnop

X jΦ j(ξ∞)

Notice on the criterion of stop:The increment is tested Δn to evaluate the convergence of Newton, because the solution and theincrement vary on the scale of the unit. The value of the increment compared to the unit is thus a

robust indicator of convergence. If the criterion of stop related to the value of the function lsnh (ξn) , it

would be necessary to put on the scale the result. Very concretely, when the increment tends towards zero compared to the unit, the solutiontn+1=tn−Δn becomes stationary compared to the unit. From a digital point of view, that guarantees

convergence on the interval [0 ,1] , taking into account the precision machine.

By way of an example, now let us consider the interpolation of a level-set in the quadratic element(QUAD8) paragraph §2.2.3. Let us consider a segment AB arbitrary in the element relative (to seeFigure 3.3.8.1-3). By tracing the evolution of the level-set interpolated f : t→ lsnh(t ) along thesegment A -B (to see Figure 3.3.8.1-4), it is noted that the level-set varies in a strictly monotonousway in the vicinity of the point of intersection P . As long as one remains in the vicinity of the point P ,the algorithm of Newton will observe the conditions of optimal convergence.




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The algorithm of Newton suggested above, is initialized by a linear interpolation of the level-set alongthe segment A -B Figure 3.3.8.1-5. The point of initialization P0 is then sufficiently close to the

convergence point P . The algorithm of Newton will have an optimal behavior.

On the other hand, if the algorithm had been initialized at the point A , proximity of a point ofstationnarity (where the derivative first of f : t→ lsnh(t ) cancel yourself) would have caused thefailure of Newton. Several scenarios would be possible:

• that is to say Newton would not converge at the end of a maximum iteration count given,• that is to say Newton would leave the segment A -B .

The presence of the point of stationnarity has a geometrical justification: the Iso-zero of the level-set ispractically parallel to the segment A -B in the vicinity of the point A (see Figure 3.3.8.1-3).Thereafter, the distance enters the segment A -B and the Iso-zero decreases gradually.Consequently, in the vicinity of the point of intersection variation of the level-set becomes again strictlymonotonous.

To prevent this configuration, Newton is initialized sufficiently near to the point of intersection Figure3.3.8.1-5. Moreover, with each iteration of the algorithm, one controls the persistence of the solutionon the segment A -B i.e. ∀ t , t∈[0 ,1] . That brings one then additional guarantee on therobustness of the calculated solution, by Newton.


Figure 3.3.8.1-3 : research of the point of intersection enters theIso-zero of the level-set and a segment A-B (arbitrary) in an

element QUAD8



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3.3.8.2 Calculation of the points medium

At the time of the cutting of a quadratic tetrahedron, the new points mediums can be gatheredaccording to 3 categories Figure 3.3.8.2-1 :

• the points mediums of first type MP 1 : located on the edges of the tetrahedron (triangle),between the node tops and points of intersection,

• the points mediums of second type MP 2 : located on the Iso-zero, in the quadratic face (SE3,TRI6 or QUAD8) delimited by the points of intersection. The points mediums on the edges of


Figure 3.3.8.1-4 : evolution of function outdistances (level-set) alongsegment A-B

Figure 3.3.8.1-5 : Initialization of the algorithm of Newton-Raphson by linearinterpolation of the level-set



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the face and the point medium in the center of the face (if face quadrangle) are calculated withslightly different procedures,

• the points mediums of third type MP 3 : located on the faces quadrangles of the under-polyhedron-children (pentahedron or pyramid), generated by the cutting of the tetrahedron.

Definition of the points mediums

One proposes to define systematically the points mediums in the reference mark of reference of theelement relative. As the element of reference is on right board and plane faces, that facilitates largely thecalculation of the points mediums in particular the nodes mediums of first type. Let us underline ofadvantage this point essential, calculations to come will be based often implicitly on the assumption thatthe faces of the tetrahedrons to be cut out are plane in the element of reference relative.However, with this definition, it is possible to calculate a node medium which is not “with medium “of theedge in the real frame of reference. That is not awkward, on condition that choosing a technique ofadapted integration.

In the element of reference, we propose the following definition of the points mediums:

• for a point medium of first type , located between Nœud top XX and the point of intersection

IP : ξPM1=ξMilieu(XX - IP )=

ξXX +ξ IP2

As represented on Figure 3.3.8.2-3 .


Figure 3.3.8.2-1 : various types of nodes mediums MPi tocalculate at the time of the cutting of the quadratic elements



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Exception: For the quadratic elements multi-fissured, in the element of reference relative, theintersected edge and on which one wishes to position the points mediums is not necessarilyright. If the intersected edge is the side of under element of integration generated by cutting inaccordance with a first crack (see Figure 3.3.8.2-3 ), it is necessary to take into account thepossible curve of the edge to position the points mediums of first type:

ξM 6=ξI 4

+ξ I2

2

M 6=ξM 6

(ξM 6−1)

2I 2+

ξM 6(ξM 6

+1)

2I 3+(1−ξM 6

)(1+ξM 6) I 4


Figure 3.3.8.2-3 : determination of the points mediums M 6 and M 7 on the edge I2 I 3

Figure 3.3.8.2-2 : calculation in thereference mark of reference, of a

point medium of second type,located between 2 points of

intersection



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• for a point medium of second type , located between 2 points of intersection on a face of the

tetrahedron Figure 3.3.8.2-4 , the following choice is performed: ξPM 2

=ξM+t v

lsnh (ξPM 2 )=0

◦ M is the medium of the segment uniting the 2 points of intersection ξM=ξ IP1

+ξIP2

2

◦ v is the vector unit normal with IP1 - IP2 , pertaining to the face (A ,B ,C ) (planes in

the element of reference). Concretely to calculate the normal vector v , the followingequations are posed:

{v ∈ (A , B ,C )

v ⊥ IP1 - IP2

∥v∥ = 1 }⇒{v = α1 AB + α2 ACv ⋅ IP1 - IP2 = 0

∥v∥ = 1 } After resolution:

v = ±α1 (k2 AB − k1 AC ) with

α1 = 1/√n12k 2

2+n22k 1

2−2k k 1 k 2

n1 = ∥AB∥ et n2 = ∥AC∥

k 1= AB ⋅ IP1 - IP 2

k 2=AC ⋅ IP1 - IP 2

k= AB ⋅ AC

Note:

1. If k2=0 i.e. AC ⊥ IP1 - IP2 , then, v = ±AC

∥AC∥

2. There are 2 possible vectors directing unit for a given line, which explains the 2solutions calculated for the vector v . One chooses one of the 2 solutionsarbitrarily. This choice does not have any influence on the convergence of thealgorithm of Newton.


Figure 3.3.8.2-4 : calculation in thereference mark of reference, of a

point medium of second type,located between 2 points of

intersection



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• for a point medium of second type , located at the center of a face quadrangle, the followingchoice is performed:

ξPM 2=ξM+t v

lsnh (ξPM 2 )=0

◦ M is the point medium of the edge [ IP1 - IP4 ] i.e. ξM=ξ IP1+ξIP4

2 . If the Iso-zero is

plane cutting generates under-tetrahedrons on right board then.

◦ v is the gradient of the lsn at the point M: v=grad lsn(M ) . It is “the best” direction ofresearch because the gradient of the lsn is by definition orthogonal on the surface lsn=0that one seeks to reach.

◦ Calculation of the points mediums of second type:The points mediums of second type check the equation of intersection of a line with the Iso-zero following:

ξPM 2=ξM+t v

lsnh (ξPM 2 )=0

According to the localization of the point medium (either located between 2 points of intersection, orlocated inside the face quadrangle on the Iso-zero), we clarified the exact coordinates of ξM and vin the element of reference.

One solves the research of the zero of the equation lsnh (ξM +t v )=0 by a Newton as in theparagraph §2.2.3. In order to make sure that the algorithm of Newton “remains” respectively in the faceSORTED for a point medium of second type located between two points of intersection and underelement TETRA for a point medium of second type located at the center of a face quadrangle, onereinforces the algorithm of Newton by the method of Dekker. If an iteration of the algorithm of Newtonsends to us apart from the zone of authorized research, a dichotomy will be preferred to approach the

solution. Indeed, it happens that the equation ξPM 2

=ξM+t v

lsnh (ξPM 2 )=0 have several solutions, but there is of

them only one which is in the face SORTED of research or under element TETRA of research. Thissolution can prove to be difficult to collect by the algorithm of Newton, especially when the curvelsn=0 has an important curve or “shaves” an edge of the face SORTED or a face of under elementTETRA (see Figure 3.3.8.2-6 and Figure 3.3.8.2-7).


Figure 3.3.8.2-5 : calculation in the reference markof reference, of a point medium of second type,

located on a face quadrangle



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Figure 3.3.8.2-6 : existence of solutions brought closer for the point medium of the type 2 in a faceSORTED

Figure 3.3.8.2-7 : existence of solutions brought closer for the point medium of the type 2 locatedon a face quadrangle under element TETRA



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Algorithm of Newton-Raphson for the calculation of a point medium located on the isozérolevel-set:

▪ Definition of ξM starting point of the search and for v direction of research.

▪ Initialization of Newton - Raphson: t0=0

▪ Determination lower and higher limits tmin and tmax space of research (see Figure3.3.8.2-8 and Figure 3.3.8.2-9)

▪ As long as n⩽itermax to make:

• ξn=tn v+ξM 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 2018 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)

Figure 3.3.8.2-9 : limits of “the space of research” under elementTETRA for a point medium of second type located at the center of

a face quadrangle

Figure 3.3.8.2-8 : limits of “the space of research” in the faceSORTED for a point medium of second type located between two

points of intersection



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• Calculation of the function: lsnh (ξn)= ∑

1⩽ j⩽nnop

lsn jΦ j (ξn)

• Calculation of the derivative: dlsnh(ξn)/dt= ∑

1⩽ j⩽nnop

lsn j∇Φ j(ξn)⋅v

• Checking of the slope: if ∣dlsnh (ξn)/dt∣<10−16 then left with code error.

• Calculation of the increment: Δn=lsnh(ξn)

dlsnh(ξn)/dt

• Calculation of the new position: t n+1=t n−Δn

• If t n+1∈[ tmin ,tmax] , then one continues, if not:

• If t n+1>tmax , then t n+1=tn+tmax

2

• If t n+1<tmin , then t n+1=tn+tmin

2

• Criterion of stop on the increment: if ∣Δn∣<10−8 then fine of the loop.

▪ Calculation of the coordinates of reference of the point of intersection: ξ∞=t∞ v+M

▪ Checking of the assumption ξ∞∈ELREFP : if ξ∞∉ELREFP then left with codeerror.

▪ Calculation of the real coordinates by interpolation of the geometrical coordinates:

X (ξ∞)= ∑1⩽ j⩽nnop

X jΦ j (ξ∞)

For the nonsimpliciaux elements, there exists a situation for which the research of the point medium ofthe type 2 vain within the space of research is authorized (between the terminals inf and sup ). Asindicated in the paragraph § 3.3.1 , cutting under elements is based on a reduction of thenonsimpliciaux elements in simpliciaux elements (SORTED in 2D and TETRA in 3D). However thisreduction can bring situations which return the research of the points mediums of the type 2 impossiblein the subdivision of the nonsimpliciaux elements. An example is presented on the figure Figure3.3.8.2-11 on the left. One searches the point medium of the type 2 between the points of intersectionIP1 and IP2 . However in the interval of research authorized (between the terminals inf and sup ), the field of lsn is of constant sign. The algorithm of Newton-Raphson is thus put in failure. Thediagonal edge interns is indeed intersected by the curve lsn=0 whereas the latter was detected likenot crossed because the value of lsn in each one of its 3 nodes is strictly of the same sign. But unlikean edge of the element relative, the field lsn along this internal edge depends on the value of lsn ineach of the 8 nodes of the element relative QUAD8, and not only of the value of lsn in its 3 nodes.For this reason the field of lsn change sign along this internal edge. This difficulty can becircumvented by choosing another bijection for the division of the quadrangle in two triangles (conferFigure 3.3-1). On Figure 3.3.8.2-11 on the right, one uses the other bijection for the cutting of thequadrangle in two triangles so as to avoid a change of sign of the field lsn along the internal edge.One then determines the points mediums of second type M 1 and M 2 in the authorized intervals.

Thus each time the search for a point medium of second type in the interval [inf , sup ] is put infailure within an element simplicial, one starts again the procedure of cutting of the element bychoosing Udifferent bijection for division under cell triangular (or tetrahedral in 3D). If all the possiblebijections would not have not allowed to bring a solution to this problem, ON then carries out a linear




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approximation of the curve lsn=0 , where the point medium is selected as being the medium of thesegment [ IP1 IP2 ] within the space of reference.

These cases are rare and occur only when the curve of the interface is important in comparison withthe size of the meshs. By refining the grid sufficiently, one is ensured not to encounter this problem.

• for a point medium of third type , located on a face quadrangle of a polyhedron-child (pyramid

or pentahedron), the virtual edge of cutting connects the first point of intersection IP1 face and

a node top of the tetrahedron B (see Figure 3.3.8.2-11 ). Concerning the positioning of thepoint medium on the face quadrangle:

◦ if the level-set is right, the following choice is performed: ξPM 3=ξMilieu( IP1 -B)=

ξ IP1+ξB2

so

that cutting generates under-tetrahedrons on right board.

◦ if the level-set is curved, the following choice is performed: ξPM 3=ξMilieu (PM 2 -D )=

ξ PM 2+ξD

2so that the point calculated medium did not leave the face quadrangle Figure 3.3.8.2-11 .


Figure 3.3.8.2-10 : case of abortion of the search for a point medium of the type 2 for thenonsimpliciaux elements.



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Notice on the criterion of curve

To judge curve of the lsn, the following step is taken. One considers initially the point medium PM3

candidate such as ξPM 3=ξMilieu ( IP1 -B)=

ξ IP1+ξB2

. One then calculates the sine of the angle between

the vectors IP1,PM 2 and IP1,PM3 , the plan being directed compared to the outgoing normalcoming towards the reader. If it is negative, this point medium will give rise to distorted elements Figure3.3.8.2-12, if not both SORTED resulting will be healthy. In practice, it is considered that the lsn is toocurved when the sine is lower than 0.1. One chooses then PM3 such as .

Under-cutting and error of discretization

In the procedure of cutting explained in the paragraph §3.3.8, the levet-set is interpolatedsystematically in the element of reference relative. Consequently, the points of intersection and points


Figure 3.3.8.2-11 : calculation in thereference mark of reference, of a pointmedium of third type for a curved Iso-

zero of form

Figure 3.3.8.2-12 : criterion to determine if the lsn is considered to be curved: situation“curves” on the left and “healthy” on the right



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mediums positioned on the Iso-zero, check the equation rigorously lsnh (ξELREFP)=0 . In clearly, thealgorithm of positioning of the points on the surface of the Iso-zero does not introduce an additionalerror compared to the interpolation error of the level-set.

On the other hand, the geometrical form of the level-set is approximated by connecting the pointscalculated to form the border of the under-fields of integration (see Figure 3.3.8.2-13, where thefunction Φh= lsnh (ξELREFP)=0 is conical whereas the edge of the quadratic under-cell coinciding aswell as possible with this conical is parabolic).

3.4 Recovery of the facets of contact3.4.1 Elements XH

The recovery of the facets of contact is useful at the same time for the contact but also to apply amechanical pressure in the lips of cracks. To obtain these facets, one is based on the work of cuttingunder elements describes in §3.3. Indeed, the topology of under elements of integration containsalready all information on the facets of contact, which are not other than the sides (in 2D) or the faces(in 3D) of under elements of integration which check lsn=0 . It is thus enough simply to traverse thesides or the faces of under elements of integration and to select those which are confused withdiscontinuity (see Figure 3.4.1-1). However the nodes tops of under elements of integration which arepoints of intersection are located by their connectivity ranging between 1000 and 1999. In the case of apoint of intersection confused with a node of the element relative, one will have directly lsn=0 forthis node. The facets which one recovers are SEG2 in 2D linear, SEG3 in 2D quadratic, TRIA3 in linear 3D andTRIA6 in quadratic 3D. In order to recover the facets only one and only once, one carries out researchonly on the sides or the faces of under elements of integration such as lsn<0 .


Figure 3.3.8.2-13 : approximation of the Iso-zero



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In the example above, it QUAD8 is divided into 6 TRIA6. One carries out the research of the facets ofcontact on under elements 2.3 and 4 which check lsn<0 . Two then are recovered SEG3 (in red).

Algorithm of recovery of the facets of contact for elements XH:

• Buckle on nse under elements of the element relative XH

◦ If under element checks lsn<0 , then

▪ Buckle on the sides (in 2D) or the faces (in 3D) of the subelement

▪ Buckle on the nodes tops on the side or the current face

- if the connectivity of the node lies between 1000 and 1999, it is a point ofintersection with discontinuity, one selects it.

- if the connectivity of the node top is strictly lower than 1000, then it is anode top of the element relative, one selects it only if it checks lsn=0 .

▪ End of the loop on the nodes tops of the current face

▪ If all the nodes tops on the side or the current face were selected, it isabout a facet of contact, which one stores the connectivity and thecoordinates of his nodes

▪ End of the loop on the sides (in 2D) or the faces (in 3D) of the subelement

◦ End of the condition


Figure 3.4.1-1 : under elements of integration and facets of contact in a QUAD8



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• End of the loop on nse under elements of the element relative XH

3.4.2 Elements XHT and XT

When the element is XHT or XT, an additional work is necessary compared to the elements XH.Indeed, cutting in subelements is carried out in accordance with lsn , but not in accordance with lst .The facets of contact which one recovers thus check well lsn=0 , but not necessarily lst<0 ,because elements XHT and XT are likely to contain the bottom of crack. A recutting conforms to lstis then necessary.

Once obtained facets of contact such as lsn=0 , one them redécoupe so necessary according to thecurve lst=0 (see Figure 3.4.2-1)

With this intention, one uses exactly the same approach as at the time of the cutting of the principalelements under elements of integration (cutting in conformity with surface lsn=0 ). In the examplebelow, it SEG2 of right-hand side is cut out according to the curve lst=0 .

3.4.3 The multi-Heaviside elements

For the multi-fissured elements, the procedure is the same one. Indeed, cutting under elements iscarried out in a sequential way, crack after crack, so that the final subelements of integration are inconformity with each crack (see Figure 3.4.3-1). One then uses, for each crack, the same process asfor the elements XH. However, for the cracks on which another crack connects, one does not carry outnecessarily the research of the facets of contact in the elements such as lsn<0 . In order to makesure that the facets of contact are in conformity with the junction, one carries out the research of thefacets of contact on the side where the crack connects (see Figure 3.4.3-1).


Figure 3.4.2-1 : recutting of the facets on the level of the bottom of crack



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When several cracks connect on the same crack within an element, if all the connections are notmade a same side, it does not exist in general of facets of contact in conformity with each crack(seeFigure 3.4.3-2). The algorithm of cutting then transmits a message of alarm which invites the user torefine the grid so as to more not have multiple junctions within the same element. There is no recutting necessary because the multi-fissured elements cannot contain basic of crack,they are elements of interface only.


Figure 3.4.3-1 : recovery of the facets of contact for an element multi-fissured. The facet of crack 1 arerecovered side or crack 2 connects.



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Algorithm of recovery of the facets of contact for the multi-fissured elements:• Buckle on nfiss cracks which cross the element

signe ref=−1 If one or more cracks connect on ifiss , one determines the sign corresponding to the sidewhere the cracks connect signe ref=signe(côté) . If the cracks do not connect all on thesame side, one transmits the message of alarm according to:

“Several cracks are connected on both sides of the same crack within the same element. Therecovery of facets of contact in conformity with each junction is likely to fail. Refine the grid inorder to that this situation does not arrive any more”

• Buckle on nse under elements of the element relative XH

◦ If under element checks signe sousélément (ifiss)=signe ref , then

▪ Buckle on the sides (in 2D) or the faces (in 3D) of under elements

▪ Buckle on the nodes tops on the side or the current face

- if the connectivity of the node lies between 1000 and 1999, it is a point ofintersection with one of the cracks, one preselects it. It definitively isselected if ∣lsnifiss∣<loncar∗10−8 with loncar length characteristic ofthe element relative. This part of the programming will evolve in order toget rid of this criterion. Indeed, two nodes on the same side,connectivities understood enters 1000 and 1999 carry out the minimumof only one level set which it is thus easy to identify;


Figure 3.4.3-2 : recovery of the facets of impossible contact for a multi-fissured element in which

two cracks connect on both sides of a first crack.



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- if the connectivity of the node top is strictly lower than 1000, then it is a

node top of the element relative, one selects it only if it checks lsnifiss=0.

▪ End of the loop on the nodes tops of the current face

▪ If all the nodes tops on the side or the current face were selected, it is about a facet ofcontact for the crack ifiss , of which one stores the connectivity and the coordinatesof his nodes

▪ End of the loop on the sides (in 2D) or the faces (in 3D) of under elements

◦ End of the condition

• End of the loop on nse subelements of the element relative XH

End of the loop on nfiss cracks which cross the element

3.4.4 Maximum number of recovered facets3.4.4.1 Case 2D

In the case 2D, one can have to the maximum 3 facets of contact in a quadrangle (to see Figure3.4.4.1-1).

3.4.4.2 Case 3D

In the case 3D, for a hexahedron, one can raise the maximum number of facets of contact for only onefissures by 18. One is indeed likely to reach this figure for an element which contains the bottom ofcrack. During the recutting of the triangular facets in accordance with lst , certain facets give rise totwo facets ( to see Figure 3.4.4.2-1 ) , which explains why one can have with final close to 18 facets.


Figure 3.4.4.1-1 : a quadrangle with 8 nodes and its 3 facets of contact



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Figure 3.4.4.2-1 : recutting of a triangular facet on the level of the bottom of crack



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3.4.4.3 Case multi-Heaviside

In the multi-fissured case, the maximum number of facets of contact by crack is higher than themaximum number of facets of contact for an element crossed by only one fissures. Indeed, sequentialcutting generates more under elements for each new crack and thus more facets of contact becausethese last rest on under elements. In 3D, for a hexahedron, one fixes the maximum number of facets ofcontact by cracks at 30. An element 3D crossed by 4 cracks can thus have a total of 30∗4=120facets. In 2D, one fixes the maximum number of facets of contact by cracks at 7. An element 2Dcrossed by 4 cracks can thus have a total of 4∗7=28 facets.

3.5 Integration rigidity3.5.1 Intégrande of the term of rigidity mechanics

Approximation X-FEM of the field of displacement is written in vectorial form:

{uh }={ϕi ϕ jH j ϕk F1 ϕ k F

2 ϕ kF3 ϕk F

4 }⋅{a ib jck

1

ck2

ck3

ck4}

That is to say still:

{uh }={N }⋅{u } where {N } is the vector of the base enriched by the functions of form and {u } the vector of the nodalddls of displacement.The deformation are written:

{ ε }=[B ]⋅{u } where [B ] is the matrix of the derivative of the functions of forms

[B ]={(ϕ )i′

(ϕ j )′ H j (ϕk )

′ F α+ϕk (F α )′

} , α=1,4

On a finite element X-FEM occupying a field e , the form of the elementary matrix of rigidity is(according to the expression éq 3.1.4 of the PTV):

[ K e ]=∫ e[B ]t [ D ] [B ] d e

One subdivides the integral in a sum of integrals on the subelements:

[Ke ]= ∑sous-éléments

∫ se[B ]t [D ] [B ] dse

Let us note on the way that the quantities to be integrated do not change, and always refer to the finiteelement X-FEM e (called “element relative”) and not with the subelements se . On each subelement, the intégrande is continuous. Indeed, the subelements respect the discontinuity

of the crack, therefore the function H j ( x ) there is constant, and the functions i ′ , F and F

′

there are continuous.

3.5.2 Intégrande of the geometrical term of rigidity

When one places oneself within the framework of great rotations or the great deformations, it isnecessary to take into account geometrical rigidity. Geometrical rigidity is also useful in the case of themodal analysis.




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In the case of great rotations for example, the virtual work of the internal forces can be written on theinitial configuration in the form:

int=∫S E : E∗ d

where E and S are the vectors of deformation of Green-Lagrange and Piola-Kirchhoff constraint ofsecond species respectively.The iterative variation of virtual work is written then:

int=∫ SE :E∗

SE : E∗ d

and by using the fact that:

[E ]=12

Ñu+Ñ uT +Ñ uT Ñu

[ E ]=12

∇ u∇ uT∇ uT∇ uT∇ uT ∇ u

and:

[ E ]=12

∇ uT ∇ u∇ uT ∇ u

variation virtual work becomes:

int=∫ S E :E∗

tr ∇ T u∗⋅S⋅∇ δu d

The integral on an element X-FEM of the second term of this equation which is written (in 2D):

∫e

tr ∇T u∗⋅S⋅∇ δu d se=∑se∫

se∑m, n

aX∗ aY

∗ bX∗ bY

∗ cX∗ cY

∗ m⋅[K segeom]m, n⋅

δaX

δaY

δbX

δbY

δcX

δcYn

dse

fact of appearing the geometrical matrix of rigidity of the subelement:

[K segeom]m, n=

, im

Sij, jn

0 H,im

S ij, jn

0 ,im

Sij , jn

F j

nF, j

0

0 ,im S ij, j

n 0 H,im S ij, j

n 0 ,im S ij , j

n F jn F, j

H ,im Sij, j

n 0 H 2,im S ij, j

n 0 H ,im S ij, j

n F jn F , j

0

0 H , im S ij, j

n 0 H 2,im S ij, j

n 0 H , im Sij , j

n F jn F , j

, im F

im F, i

S ij, j

n0 ,i

mF i

m F, iSij H , j

n0 , i

m Fi

m F ,iS ij, j

n F j

nF , j 0

0 , im F

imF ,i

S ij, j

n 0 ,imF

im F ,i

S ij H , j

n 0 ,imF

im F, i

Sij , j

n Fj

n F , j

where n and m correspond to the numbers of the nodes in the element relative.

3.5.3 Calculation of the derivative of the singular functions:

One seeks the derivative of the singular functions compared to the total coordinates x , y , z . Thesingular functions are given according to the polar coordinates r ,θ .




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Derived from the singular functions in the polar base r ,θ :

[∂F 1

∂ r∂ F1

∂θ∂ F 2

∂ r∂F 2

∂θ∂F 3

∂ r∂ F3

∂θ∂ F 4

∂ r∂F 4

∂θ

]=[1

2rsinθ2

r2

cosθ2

12r

cosθ2

− r2

sinθ2

12r

sinθ2

sin θ r2

cosθ2

sin θr sinθ2

cosθ

12r

cosθ2

sin θ − r2

sinθ2

sin θr cosθ2

cosθ]

Derived from the singular functions in the local base {e1 , e2 , e3 } :

[∂F 1

∂ x1

∂ F1

∂ x 2

∂ F 1

∂ x3

∂F 2

∂ x1

∂F 2

∂ x 2

∂ F 2

∂ x3

∂F 3

∂ x1

∂ F3

∂ x 2

∂ F 3

∂ x3

∂F 4

∂ x1

∂F 4

∂ x 2

∂ F 4

∂ x3

]=[∂ F1

∂ r∂ F1

∂θ∂F 2

∂ x1

∂ F2

∂θ

∂F 3

∂ x1

∂ F3

∂θ

∂F 4

∂ x1

∂ F4

∂θ

] [ cos θ sin θ 0−sin θr

cosθr

0 ]

Derived from the singular functions in the total base {E1 , E2 ,E3 } :

[∂F 1

∂ x∂ F1

∂ y∂ F 1

∂ z∂F 2

∂ x∂F 2

∂ y∂ F 2

∂ z∂F 3

∂ x∂ F3

∂ y∂ F 3

∂ z∂F 4

∂ x∂F 4

∂ y∂ F 4

∂ z

]=[∂ F1

∂ x1

∂ F1

∂ x2

∂F 1

∂ x3

∂F 2

∂ x1

∂ F2

∂ x2

∂ F 2

∂ x3

∂F 3

∂ x1

∂ F3

∂ x2

∂F 3

∂ x3

∂F 4

∂ x1

∂ F4

∂ x2

∂ F 4

∂ x3

] [ P ]Ee−1

where [P ]Ee is the matrix of passage of the total base at the local base, such as

e1 e2 e3

[P]Ee = [× × ×× × ×× × ×]

E1

E2

E3

To express a quantity in the total base, one carries out the product of this matrix by the quantityexpressed in the local base, that is to say

X E=[P ]Ee X e

This matrix being orthonormal, its reverse is obtained easily:




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[P ]Ee−1=[P ]Ee

t

3.5.4 Diagrams of integration3.5.4.1 Integration of the nonsingular terms

For the nonsingular terms (classical terms and Heaviside enrichment), the functions to be integratedare polynomials. Digital integration by a diagram of Gauss-Legrendre on each under-tetrahedron thenis well adapted. That is to say npg the number of points of Gauss. These points have as coordinates

in the under-tetrahedron of reference sg , tg , ug and the associated weight is noted w g . That is to

say J Jacobienne of the transformation under-tetrahedron under-tetrahedron of reference. Itsdeterminant is constant by subelement. The classical procedure is written:

∫se

f d se=∫ seref f ∣det J ∣d se≈∑

g=1

npg

f sg , t g , u g wg∣det J ∣

The function f being only known in the reference mark related to the element relative of reference, itis necessary to express the coordinates of the point of Gauss sg , tg , ug in this reference mark[feeding-bottle33]. First of all, one transports them in the real reference mark (simple use of thefunctions of the shape of the under-tetrahedron), then one transports them in the reference markrelated to the element relative of reference (solution of non-linear equations by the algorithm ofNewton-Raphson):

sg , t g , ug t1

xg , yg , z g t2

g , g ,g

Notice :

Indeed, it would rather be necessary to write ∫ seref f °t2°t1 s g , t g , u g ∣det J ∣d se . The

transformation t1 is linear. On the other hand, the transformation t2 is non-linear. It is thereverse of a linear transformation. It is a combination of rational fractions and square roots withseveral variables. So one does not know the effect of f °t 2°t 1 on the maximum order of thestudents' rag processions in the case general. On the other hand, for a real element on parallel

board (rhombus in 2D), t2 being linear, it is considered that f °t2°t1 sg , tg , ug are students'

rag processions in , ,g g gs t u same order as those definite on the element relative of

reference.

The functions of form of the isoparametric elements are of the students' rag processions of the type

ijk with i jk≤ p (see it Table 3.5.4.1-1). The derivative of the functions of form are then of

the students' rag processions in the same way standard with i jk≤ p−1 . The quantities to be

integrated are thus of the students' rag processions of order m= p−1 2 . One refers to [feeding-bottle34] to determine the adapted diagram allowing an exact integration3 of all the terms. For theelements on nonparallel board, integration will be then approximate.

Element relative p m= p−1 2 npg

hexahedron 3 4 15pentahedron 2 1 1tetrahedron 1 0 1

Table 3.5.4.1-1 : Diagrams of integration according to the type of the element relative

3.5.4.2 Integration of the singular terms

The diagrams of digital integration of Gauss-Legendre were conceived for the integration ofpolynomials. However the presence of students' rag processions in 1/ r coming from enrichment

3 An integration preserving the order of approximation finite element would be in fact sufficientWarning : 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 2018 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



6bd3f8cf6f8b

with the asymptotic fields leads to a very slow convergence of the precision of these diagrams: aconsiderable number of points of Gauss is then necessary to obtain an acceptable error. When thesingularity is on the border of the field of integration, of the quite selected changes of variablessuccessive make it possible to be reduced to integration on a unit field of a function not-singular,polynomial (or quasi-polynomial), on which a classical integration of Gauss-Legendre quickly proves tobe effective and convergent [feeding-bottle28]. About ten points of Gauss in each direction is enough toreach the limits of the digital precision of a calculator. When the singularity is not on the border, it isadvisable to cut out the field under-fields, so that the singularity is on the borders of the under-fields. Inspite of very satisfactory results in term of convergence of the relative error of integration on each term[feeding-bottle28], the effort of implementation required for the structures 3D appears considerable tous [feeding-bottle35]. More especially as the difference on the level in the total errors (after resolution)between a classical integration with a number of points of rather high but reasonable Gauss and amodified singular integration is not really significant

For the moment, one set up the same classical diagram of Gauss-Legendre as that which is used forthe integration of the nonsingular terms.

3.6 Integration of the second surface members3.6.1 Intégrande of the second member of the surface efforts

The second members due to the surface efforts are written in a discretized way:

∫Γ tt⋅u∗dΓ t

surface Γ t discretized corresponds to the faces of the elements 3D. a possible under-cutting for itsfaces 2D is that corresponding to the trace of the under-cutting of the elements 3D. In Aster, onechooses to take generic under-cutting 2D, corresponding to that described in the paragraph [§3.3.7].

Just as for the integration of rigidity, it is necessary to determine the coordinates of the points of Gaussof the under-triangle of reference in the element relative of reference

That is to say the point of Gauss of coordinates sg , tg in the under-triangle of reference. Theproblem is that the environment of calculation is 3D, One thus creates a real reference mark 2D local

with each under-triangle Is the under-triangle ABC , of normal n=AB∧AC

∥AB∧AC∥. The local base is

written:

x l=AB

∥AB∥, y l=n∧ x l .

The point of Gauss has then as local coordinates:

x g yg =∑i=1

3

ψ i s g ,t g xi ,∑i=1

3

ψ i s g , t g yi

where ψ i are the functions of forms of the under-triangles and x i , yi coordinates of the tops of theunder-triangle in the local reference mark 2D x l , yl .As for the integration of the terms of rigidity, it is now necessary to use a method of Newton-Raphsonto determine the coordinates of the point of Gauss in the reference mark of reference of the elementrelative (element of face). For that, all the real coordinates are expressed in the real local referencemark 2D.

3.6.2 Intégrande of the second member of the surface efforts on the lips of the crack

The second members due to the surface efforts on the lips of the crack are written in a discretized way:

∫Γ cg⋅u∗dΓ

Integration is done on the two lips of the crack ( Γ c=Γ1Γ 2

), with for convention:

•the lip Γ 1 is such as the Heaviside function is negative and the polar angle is worth −π ,




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•the lip Γ 2 is such as the Heaviside function is positive and the polar angle is worth π ,

Note:

The normals external with the lips being opposite, the surface efforts equivalent to the imposedpressure have opposite signs.

The integration of the second member due to the efforts applied to the lips is done on the facets of thelips, thanks to the data of the objects corresponding to the discretization of the lips in triangular facets(see [§4.3 in R5.03.54]). These objects were created at the origin for the taking into account of thecontact.

For the elements 2D, the options CHAR_MECA_PRES_R, CHAR_MECA_PRES_F,CHAR_MECA_PCISA_R and CHAR_MECA_CISA_F allow to apply mechanical efforts of pressuredistributed and shearing to the facets of contact. Only options CHAR_MECA_PRES_R andCHAR_MECA_PRES_F are currently programmed for the elements 3D, which means that it is notpossible to apply to the lips of the shearing forces.

In multi-cracking, one can have different surface efforts on various cracks.

Figure Ci below (Figure 3.6.2-1) illustrate digital surface integration on the facets of contact of aquadrangle. In each point of Gauss of the facet, one determines a normal unit vector and a tangent unitvector only using the topology of the facets of contact.

In 3D, one determines in the same way in each point of Gauss a unit normal vector only starting fromthe topology of the facets of contact. One can possibly create an orthonormal base in each point ofGauss by supplementing this normal vector by two tangent with discontinuity and orthogonal unitvectors between them.


Figure 3.6.2-1 : surface integration on the facets of a quadrangle



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Figure 3.6.2-2 : surface integration on the facets of a tetrahedron.



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4 Linear thermics with X-FEM4.1 Introduction

Method X-FEM was introduced into the linear operator of thermics in order to be able to carry outthermomechanical calculations chained by taking of account the crack in the thermal resolution.: thefield of temperature is then discontinuous through the crack if no thermal loading is applied to its lips(with a natural condition of null flow). It is also possible to define a condition of heat exchange betweenthe lips of the crack by the data of a coefficient of exchange.

Notice :

The resolution of the problems of linear thermics with X-FEM is based on a algorithmy quasi-identicalto that put in work for the mechanical resolution of problems of structures fissured with X-FEM, detailedin the part 3 “ Problem of cracking with X-FEM ”. In particular procedures allowing:

• the enrichment of the approximation of the field of the nodal unknown factors (statute of thenodes, statute of the meshs, cancellation of the degrees of freedom nouveau riches wrongly);

• the under-cutting of the elements crossed by a crack;• procedures of integration of the matrices and the elementary vectors on this under-cutting ;

are identical to those which were previously detailed, it will thus not be it in this part.

4.2 Restrictions

The whole of modelings and the types of loading available for the algorithm of linear thermics transitoryin the case of model classics (not X-FEM) was not completely reflected for finite elements thermal X-FEM. All these features remain available of course on the part not-enriched by the model, and aredocumented in [feeding-bottle74]. Evolutions to come will make it possible to come to supplementgradually the list of the features available to date. Type of with dealt problems

Resolution of linear thermal transients (independence at the temperature of parameters materials), foran isotropic material (scalar conductivity). The connection of crack (multi-Heaviside) is excluded.

Finite elements available

For the moment, only elements X-FEM of modelings 3D, axisymmetric, and planes were introduced forthe following linear meshs support:

modeling meshs support principal elements meshs support elements of edge

3D TETRA4, HEXA8, PENTA6, PYRA5 TRIA3, QUAD4

plane/axisymmetric TRIA3, QUAD4 SEG2

There thus do not exist thermal elements X-FEM lumpés, and the grid from which the enriched modelis defined is obligatorily linear.

Thermal loadings available on the part enriched by the model

For the moment, the only loading available on the part enriched by the model is the condition of heatexchange between the lips of the crack. No other boundary condition can be imposed on thermalelements X-FEM of edge, or on nodes belonging to elements thermics X-FEM. In the same way, nosource term can be imposed on thermal elements X-FEM principal. The user must thus ensure himselfto apply his loadings sufficiently “far” crack, i.e. on elements not nouveau riches (or nodes whosesupport is made only of elements not nouveau riches).

4.3 With dealt problemWarning : 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 2018 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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In order not to weigh down the expression of the variational formulation, one considers the simplifiedproblem of a fissured solid, made up of a linear and isotropic material, of which part of the border(sufficient far away from the crack) is subjected to an imposed temperature. The rest of the border(excluded crack) is subjected to a condition of null flow. Lastly, one imposes a condition of heatexchange between the lips of the crack.

A crack is thus considered c in a field ⊂ℝndim , with ndim=2ou 3 . One notes n the outgoing

normal at its border ∂ , 1c and 2

c lips of the crack of respective outgoing normals n1 and n2 ,

and d the part of ∂ subjected to a temperature forced (see Figure 3.3.8.1-3 ). One notes x the

variable of space, t time, and T the temperature. The interval ] t 0 , t f ] corresponds to the timeinterval considered for the resolution of the problem.

Figure 4.3-1: Notations

For the problem considered, while noting respectively and C p the thermal coefficient ofconductivity and voluminal heat with constant pressure, the equation of heat is written:

−∇ . ∇ T x ,t =C p∂T∂ t

x , t , ∀x , t∈× ] t0 ,t f ] éq 4.3-1

while noting T 1=T | 1c , T 2=T | 2

c , and h the coefficient of heat exchange between the lips of thecrack, the boundary conditions are written:

{T=T sur d

∂T 1

∂ n1

=hT 2−T 1 sur 1c

∂T 2

∂ n2

=h T 1−T 2 sur 2c

éq 4.3-2

with the initial condition:

T x ,0=T 0 x , ∀ x∈

One introduces then following functional spaces:

V={v∈H 1 , v | d=T } and V 0={v∈H 1

, v | d=0}

The variational formulation associated with éq 4.3-1 and éq 4.3-2 is written then: to find T ∈V suchas




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∫

C p∂T∂ t

v d∫

∇ T .∇ v d

∫1c

h T 1−T 2v d ∫ 2c

hT 2−T 1v d =0, ∀ v∈V 0

éq 4.3-3

The discretization in time is carried out by one - diagram (see [feeding-bottle74]). One placesoneself between two stakes of successive times t n and t n+1 , one notes t=t n1−t n the step oftime, and ∈[0,1] the parameter of - diagram. The following notations then are introduced:

T +=T x , tn1 , and T

-=T x , tn

V +={v∈H 1

, v | d= T + } and V -= {v∈H 1

, v |d= T - }

The application of - diagram with éq 4.3-3 conduit with the following variational formulation: beinggiven T -

∈V - to find T +∈V + such as

∫

C pT +

tv d∫

∇ T + .∇ v d

∫1c

h+T 1

+−T 2

+v d ∫

2c

h+T 2

+−T 1

+v d

=∫

C pT -

tv d−∫

1−∇ T -.∇ v d

∫ 1c

1−h -T 2

-−T 1

-v d ∫

2c

1−h -T 1

-−T 2

-vd , ∀ v∈V 0

éq 4.3-4

4.4 Approximation X-FEM of the field of temperature

One uses the notations previously adopted for mechanics in the paragraph 3.2 “ Enrichment of theapproximation of displacement ”. The classical approximation finite elements is pointed out:

T h( x)= ∑i∈N n (x )

T iφi( x)

The enriched approximation is written as for it:

T h( x)= ∑i∈N n (x)

T iCφi( x )+ ∑

j∈N n( x)∩K

T jHφ j( x )H j (lsn( x))+ ∑

k∈N n (x)∩L

T kCTφk ( x)F

1 (lsn( x) ,lst (x ))

This expression is made up of 3 terms. The 1er term is the classical term where T iC indicate the

classical degrees of freedom, the second term is the term enriched by the function by selection by fieldH j with T j

H degrees of freedom corresponding nouveau riches, and the third term is the term

enriched by the singular function F1 (enrichment “Ace-Tip”) with T kCT degrees of freedom

corresponding nouveau riches.

It is supposed that the field is partionné Ω by the interface of the crack of the crack c in two fields

Ω- on the side of 1c and Ω+ on the side of Γ2

c .

One points out the expression below of H j :

Si x j∈Ω+ , H j(x )={ 0 si x∈Ω+

−2 si x∉Ω+

Si x j∈Ω- , H j( x)={ 0 si x∈Ω-

+2 si x∉Ω-




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One points out the expression below of F 1 :

F 1lsnx , lst x =r sin

θ2

where r , θ are the polar coordinates in the local base at the bottom of crack determined startingfrom the level-sets lsnx , lst x .

Notice :

In mechanics, four functions determined starting from the asymptotic development of the field ofdisplacement in bottom of crack are introduced into the term of singular enrichment. In thermics, oneintroduces only the function F 1 who is the only function (among the four) discontinuous through thecrack. This kind of enrichment, retained in [feeding-bottle 75 ] in the case of a condition of null flowimposed on the lips of the crack (adiabatic crack), is introduced here in the case of an adiabatic crackas in the case of a condition of exchange between the lips of the crack.

4.5 Integration of the matrices and the elementary vectors

One details in this paragraph the expression of the elementary quantities associated in each term ofthe variational formulation éq 4.3-4 linear problem of thermics discretized in time.

4.5.1 Voluminal integrals

That is to say E a thermal element X-FEM. One notes {T}ET, [N]E and [B]E respectively the

elementary vector of the nodal unknown factors (at the moment + ), the elementary matrix of thefunctions of forms and the elementary matrix of the derivative of the functions of forms. By way of anexample, one can consider an element of dimension ndim=2 ou 3 , with n nodes, and of type“Heaviside/Ace-Tip”. {T}E

T, [N]E and [B]E are written in this case:

{T}ET=[T 1

C T 1H T 1

CT ... T nC T n

H T nCT ] of size 3n

[N ]E=[N 1 H 1N 1 F1N 1 ... N n H nN n F 1N n ] of size (1,3n)

[B ]E=[∇ N 1 ∇(H1 N 1) ∇(F1 N 1) ... ∇ N n ∇ (H n N n) ∇(F1 N n)] of size (ndim ,3n)

All the voluminal quantities are integrated according to the procedure being based on cutting insubelements, and detailed in the case of mechanics in the page 89. One gives below the expression ofthese elementary quantities.

matrix of mass (option MASS_THER ):

∫E

C p

t[N ]E

T[N ]E d

matrix of rigidity (option RIGI_THER ):

∫E

[B ]ET[B ]E d

vector relating to the temporal discretization (option CHAR_THER_EVOL ):

∫E

C p

t[N ]E

T[N ]E {T

-}E d−∫

E

1− [B ]ET[B ]E {T

-}E d




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4.5.2 Surface integrals

These quantities correspond to the condition of heat exchange between the lips of the crack, and areintegrated along the lips of the crack according to the procedure being based on the discretization ofthe lips in “facets of contact”. This procedure is detailed in the case of mechanics in the page 93. The elementary matrices first of all are introduced [ N1]E and [ N2]E , definite like the matrices of the

traces of the functions of forms respectively on 1c and 2

c :

[ N1]E=[N| 1c ]E , [ N2]E=[N| 2

c ]E Notice :

From one matrix to another, the coefficients corresponding to classical enrichment are equal, and the

coefficients corresponding to Heaviside enrichment and “Ace-Tip” are connected by: H |Γ2c−H |Γ 1

c=2 ,

and F | Γ2c

1−F |Γ1

c

1=2 F |Γ1

c

1 because the polar angle is worth ± .

One introduces then the following matrices of coupling:

[ N12 ]E=[ N1]E−[ N2]E , [ N21 ]E=[ N 2]E−[ N1]E

The elementary quantities are then given by the following expressions, with Ec the whole of the

facets constituting the discretization of the crack within the element E :

matrix relating to the condition of exchange (option RIGI_THER_PARO_R ):

∫ Ec

h+[ N1]E

T[ N12 ]Ed ∫

Ec

h+[ N2]E

T[ N21]E d

vector relating to the condition of exchange (option CHAR_THER_PARO_R ):

∫ Ec

1−h -[ N1]E

T[ N 21]E {T

-}E d ∫

Ec

1−h -[ N2]E

T[ N12 ]E {T

-}E d




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5 Energy calculation of Factors of Intensity of the Constraints

This part points out the method G - theta used in Code_Aster for the calculation of the rate of refund ofenergy in linear breaking process, following a calculation classical finite elements (in 2D or 3D). Thismethod is also used for the calculation of the factors of intensity of the constraints, by using the bilinearform of g but only in 2D. Because to be able to use it in 3D, it is necessary to know the local base atthe bottom of crack in order to express the asymptotic fields according to the polar coordinates r ,θ .

Then, one presents the contribution of the level sets in classical finite elements, which was to be ableto build a local base at the bottom of crack in 3D. As one saw in the paragraph [§2.3], the gradients ofthe level sets give a local base to the bottom of crack. Moreover, the polar coordinates in this localbase are easily expressed according to the level sets (see Figure 3.2.3-1). Thus thanks to the levelsets, method G - theta was applicable to the calculation of the factors of intensity of the constraints in3D with classical finite elements. In this context, the crack is with a grid, a classical calculation iscarried out. In postprocessing, one determines the two fields level sets starting from the crack with agrid and one from of deduced a discretization from the bottom of crack. The field theta is built startingfrom this discretized bottom of crack, and the method G - theta is applied.

The last point is the calculation of the rate of refund of energy and the factors of intensity of theconstraints, following a calculation X-FEM. In this case, the crack is not with a grid, but the followedprocedure is almost the same one: the discretization of the bottom of crack is given starting from thelevel sets, a field theta is built and G - theta is applied. The only difference is at the level of digitalintegrations: with X-FEM, integrations digital require a under-cutting of the elements to be integrated. Infact, one uses under-cutting realized for the calculation of the tangent matrices and the secondmembers.

5.1 Method G-theta for calculation of G5.1.1 Relations of balance

In linear elasticity, one defines the density of free energy , in the absence of strains and of initialstresses by the positive definite quadratic shape of the components of the tensor of the deformations:

, T =12

− th :C : − th

C is the tensor of Hooke (tensor of the 4ème order), and the two points indicate the tensorial productcontracted on two indices.

The tensor of the constraints drift of the potential to give the law of state (or law of behavior ofmaterial):

=∂

∂ ,T =C : − th

The relations of balance in weak formulation are obtained by minimizing the total potential energy ofthe system:

W v =∫

v d−∫

f i vi d−∫S

g i v id

This functional calculus being minimal for the field of displacement u :

W=∫

∂

∂ ij ij d−∫

f i v i d−∫S

g i v id

=∫

ij12 v i , j v j , i d −∫

f i vi d−∫S

g i v id

=∫

ij v i , j d−∫

f i v id−∫S

g i v i d =0




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The relations of balance in weak formulation are thus written in the following way:To find u∈V such as

∫

ij v i,j d=∫

f iv i d∫S

g ivi d , ∀ v∈V 0

5.1.2 Lagrangian expression of the rate of refund of energy

By definition, the rate of refund of local energy G is defined by the opposite of the derivative of thepotential energy compared to the field:

G=−∂W∂

The method of the virtual extensions used here is a Lagrangian method of derivation of the potentialenergy. It consists with introduced transformations:

F η :P∈M=P P ∈η

who at each material point P area of reference , associate a space point M transformed field

η . These transformations representing of the propagations of the crack, should modify only the

position of the bottom of crack 0 . Fields must be tangent on the surface of the crack.That is to say n the normal on the surface of the crack, fields must check:

j n j=0 sur cr That is to say m the unit normal at the bottom of crack, located in the tangent plan of the crack.According to [feeding-bottle50], the rate of refund of energy room G is solution of the followingvariational equation:

∫0

G⋅m=G , ∀∈ éq. 5.1.2-1

where G is defined by the opposite of the derivative of the potential energy W u with balance compared to the initial evolution of the bottom of crack:

G =−dW u d ∣

=0

(particulate derivative)

To derive the potential energy compared to his support (in the vicinity of the support of reference), thetheorem of transport of Reynolds is used:For a transformation F of class C1 and a function (tensor of order 0.1 or 2) of class C1 , while

noting the Lagrangian derivative related to the variation of field, one has the following relation:

∂

∂ ∫ d ∣=0

=∫

div ∂F

∂ =0d

In fact, D years practice, the transformation F is not always of class C1 because the field is

only C1 by pieces.

Thus,

−G = ∂∂ ∫ − f i ui d−∫

S

g i ui d ∣η=0

= ∫

− f i ui⋅

− f iu i div θ d−∫S g i ui⋅

g i u i div θ −∂θ∂nk

nk d

where:




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, T =12ii

2 ij ij−3K T−T réf

Ψ ε =∂Ψ∂ εij

ε ij∂Ψ∂T

T=σ ij εij∂Ψ∂T

T

thus:

−G =∫

ij ij∂∂T

T− f i ui− f i ui − f iui k , k d

−∫S

g iu i−g i ui g iu ik , k−∂

∂ nknk d

Firstly, let us use the relation giving the Lagrangian derivative of a field according to its derivative

eulérienne ∂

∂ :

=∂

∂ ∇⋅

According to this relation, the voluminal force f being supposed independent of , i.e. being the

restriction on fields defined on ℝ3 , one can write that

∂ f i∂ η

=0 . As it is the same for g and T

, there are the following relations:T=T , kkf i= f i , k kg i=g i , k k

Secondly, let us use the relation giving the Lagrangian derivative of the gradient of a field according tothe gradient of the field and gradient of the derivative of the field:

∇ =∇ −∇⋅∇ maybe in indicielles notations,

i , j⋅

= i , j−i , p p , j

thus

i , j=12 ui , ju j , i −

12 ui , p p , ju j , p p ,i

One replaces the 2 expressions in the 1er and one obtains:

−G =∫

ij ui , j∂∂T

T , k k− ijui , p p , j− f i , k k ui− f i ui − f i ui θ k , k d

−∫S

g i , k k ui−g i u i g i ui k , k−∂

∂ nknk d

One can eliminate the terms in u by noticing that the field u is kinematically acceptable and satisfiesthe equilibrium equation:

∫

ij u i , jd =∫

f i u id∫S

g i u id

What gives the final expression of J :




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G =∫

ij ui , p p , j − k , k∂∂T

T , k k d

∫

f i , kk u i f i ui k , k d

∫S

g i , k k uig i u i k , k−∂

∂nknk d

éq. 5.1.2-2

Notice :

This form of the integral J do not use integrals of contour but integrals of fields.

5.1.3 Discretizations

One notes s the curvilinear X-coordinate in bottom of crack. The calculation of the local value of Grequire to solve the following variational equation for several fields

i :

∫ 0G s i s ⋅m s ds=G i ∀ i∈[ 1,P ]

The scalar field G s is discretized on a noted basis p j s 1≤ j≤N :

G s =∑j=1

N

G j p j s

In the same way, fields i are discretized on a noted basis qk s 1≤k≤M . That is to say i the trace

of the field i on the bottom of crack 0 : i s=∣ 0

s and is ki components of

i s in this

base:

i s=∑k=1

M

ki qk s éq. 5.1.3-1

While injecting these expressions in the variational equation, it comes:

∫0

∑j=1

N

G j p j s∑k=1

M

ki qk s ⋅m sds=G

i , ∀ i∈[1, P ]

G j can thus be given by solving the linear system with P equations and N unknown factors:

{∑j=1

N

a ijG j=bi , i=1, P

avec a ij=∑k=1

M

ki∫ 0

p j sqk s⋅m s ds

bi=Gi

This system has a solution if one chooses P fields i independent such as: P≥N and if M≥N .It can comprise more equations than unknown factors, in which case it is solved within the meaning ofleast squares.

5.2 Method G-theta for calculation of KI, KII and KIII with the level sets

This paragraph presents the contribution of the level sets for the calculation of the factors of intensity ofthe constraints in 3D. It is based on the separation of the mixed modes thanks to the bilinear form of




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g . This bilinear form is used in Code_Aster for calculation by the method G - theta of K I , K II in2D [feeding-bottle49].

5.2.1 Bilinear form of G

The calculation of the factors of intensity of the constraints can also be done by the method theta, bydefining a symmetrical bilinear form g with:

g u , v =14

G uv −Gu−v éq. 5.2.1-1

Thus, are two fields of displacements u and v , the calculation of g u , v be carried out thanks tothe expression of G data by the equation [éq. 5.1.2-1], in which deformations related to u and vderive from the fields u and v , and constraints related to u and v are deduced by the law fromcomportemenT:

u =∇ su , v =∇ sv

u =C : u , v =C : v

By taking again the notations of [feeding-bottle49], the classical term of J θ [éq. 5.1.2-1] is:TCLA= u : ∇ u∇ − u div

= S2−S2TH −12

S1−S1TH div

It is pointed out that in elasticity, the free energy is expressed according to the coefficients of Lamé and :

=12 ii

2 ij ij− th

=12 [2 11

2 222 33

2 2 11 22113322 33 2 2122 213

2 2232 ]− th

with th=3K T−T ref tr

For each term S1 and S2 , one initially will write their usual expression, then, one will pass to thebilinear form.

5.2.1.1 Expression of S1 and S1TH One writes S1 with the following expression:

S1=C1 .S 11C2 . S 12C3 . S 13 while posing:

{C1=2C2=C3=

and

S 11=u1,12u2,2

2u3,3

2

S 12=2 u1,1 u2,2u1,1u3,3u2,2 u3,3 S 13=u1,2u2,1

2u1,3u3,1

2u2,3u3,2

2

Now let us write the associated bilinear form. The free energy depends then on two fields u and v .The coefficients materials C1 , C2 and C3 are obviously unchanged, and the new coefficientsS 11 , S 12 and S 13 have as expressions:




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S 11=u1,1v1,1u2,2v 2,2u3,3 v3,3

S 12=u1,1v 2,2u2,2v1,1 u1,1 v3,3u3,3 v1,1u2,2 u3,3u3,3v 2,2S 13=u1,2u2,1 v1,2v2,1 u1,3u3,1 v1,3v 3,1 u2,3u3,2 v2,3v3,2

The expression of the thermal part is:

SITH=3K α T u−T réf tr ε v T v−T réf tr ε u

Pour the calculation of K i , v is the asymptotic field in mode I, obtained for T v=T réf , therefore

terms in T v−T réf are simplified:SITH=3K α T u−T réf tr ε v

5.2.1.2 Expression of S2 and S2TH

As for S1 , one writes S2 with the following expression:S2=C1 . S 21C2 .S 22C3 .S 23

The coefficients materials C1 , C2 and C3 are those definite in the preceding paragraph.Coefficients S 21 , S 22 and S 23 have as expressions:

S21=∑k=1

3

∑p=1

3

uk , k uk , p p , k

S22=∑k=1

3

∑l≠k∑p=1

3

ul , l uk , p p ,k

S23=∑k=1

3

∑l≠k∑m≠ km≠l

∑p=1

3

ul , mul , p p , mul , mum , p p ,l

This writing easily makes it possible to pass to the associated bilinear form. It is enough to replace the

terms u i , juk , l by 12 ui , jv k , luk ,l v i , j . If the following notation is introduced:

B ui , j , v k , l =12 u i , j vk ,luk , l v i , j

then new coefficients S 21 , S 22 and S 23 have as expressions:

S21=∑k=1

3

∑p=1

3

B uk , k , vk , p p , k

S22=∑k=1

3

∑l≠k∑p=1

3

B u l ,l , vk , p p , k

S23=∑k=1

3

∑l≠k∑m≠ km≠l

∑p=1

3

B ul , m , v l , p p ,mB u l ,m , vm , p p ,l

The expression of the thermal part is:

S2TH=TH1

23K T u−T ref ∂ v i∂ x j

∂ j∂ x i

TH12

3K T v−T ref ∂ u i∂ x j

∂ j∂ x i

where TH1=1 in D_PLAN, AXIS and in 3D and TH1=1−21−

in C_PLAN.




6bd3f8cf6f8b

Pour the calculation of Ki , v is the asymptotic field in mode i , obtained for T v=T réf , therefore

terms in T v−T réf are simplified:

S2TH=TH1

23K T u−T réf ∂ v i∂ x j

∂ j∂ x i

5.2.1.3 Surface term

The additional term in the expression of G , had with the imposition of a surface force g on c ofexternal normal n is the following:

TSUR=∇ g⋅ ug⋅u div −n∂

∂ n =g i , kk uig i⋅ui k , k−nk∂ θ∂ nk

The term n∂

∂ n because the gradient of the field is null is orthogonal with n .

It thus remains:TSUR=∇ g⋅ ug⋅u div

=g i , kk uig i⋅uik , k

The bilinear form g u , v associated for the calculation of G and of K I , K II and K III is:

TSUR u , v = 12 [ ∇ gu⋅ vg u⋅v div ∇ gv⋅θ ugv⋅u div ]

Pour the calculation of K i , v is the asymptotic field in mode i , obtained for gv=0 , therefore termsin g v and ∇ gv are simplified:

TSUR u , v =12 [ ∇ g u⋅vgu⋅v div ]

5.2.1.4 Thermal term The additional term in the expression of G , had with the field of temperature T is the following:

TTHE=∂

∂TT ,k k =

12

3K tr v ∂T u

∂ xx

∂T u∂ y

y∂T u

∂ zz

12

3K tr u∂T v∂ x x

∂T v∂ y

y∂T v∂ z

z

For the calculation of K i , v is the asymptotic field in mode I, obtained for T v=T réf=cste ,

therefore terms in TTHE=−∂

∂T∇ T⋅=1

23K tr ∂T∂ x x∂T

∂ y y are simplified:

TTHE=12

3K tr v ∂T u

∂ xx

∂T u

∂ y y

∂T u

∂ zz




6bd3f8cf6f8b

5.2.2 Separation of the mixed modes The symmetrical bilinear form g u , v like property interesting that has to separate the three modesfrom opening of the crack. Indeed, in the form g u , v , if the field on the left is the field of solutiondisplacement and if the term on the right is a field of singular displacement in bottom of crack u I

S,

u IIS or u III

S, the factors of intensity of the constraints are expressed in the following way:

K I=E

1− 2g u ,u IS éq. 5.2.2-1

K II=E

1−2 g u ,u IIS éq. 5.2.2-2

K III=2 g u ,u IIIS éq. 5.2.2-3

Certain authors use the concept of integrals of interactions, instead of the bilinear form of g . Thesetwo vocabularies indicate in fact same the quantities. Indeed, while noting I

u , uIS the integral of

interaction enters the field solution and the singular field in mode I , the expression of K I in term ofintegral of interaction [feeding-bottle53] is:

K I=E

2 1− 2 Iu ,uI

S

This expression for K I is identical to that given by the equation [éq. 5.2.2-1] except for a

multiplicative term. If one applies the method of the virtual extensions to the bilinear form of g , theequivalent expression of the variational equation [éq. 5.1.2-1] for K i buildings is:

1−2

E∫0

K I ⋅m ds=g u ,uIS , ∀∈

1− 2 E

∫0

K II θ⋅m ds=g u ,u IIS , ∀∈

12∫0

K III ⋅m ds=g u ,u IIIS θ , ∀∈

éq. 5.2.2-4

One finds the analogue of these expressions written with integrals of interaction in [feeding-bottle54]and [feeding-bottle53]. It is pointed out that the second member of the equations [éq. 5.2.2-4] is the bilinear form of g (see[éq. 5.2.1-1]) in which the term on the left u is the field solution displacement (coming from theresolution of the problem by the finite element method) and the term on the right u i= I , II , III

S is it

asymptotic field (in mode i=I , II , III ) whose analytical expression is given by the equation [éq.3.2.3-1]. These analytical expressions are given in the local base to the bottom of crack e1 , e2 , e3exit of the gradients of the level sets, but the derivative speakers in g u ,u I , II , IIIS are madecompared to the total reference mark E1 , E2 , E3 (see the expressions of the terms S1 and S2in the preceding paragraph). It is thus necessary to carry out a basic change.




6bd3f8cf6f8b

Figure 5.2.2-1 : Polar coordinates, bases local and bases total

That is to say DPODL the matrix of the derivative of the polar coordinates in the local base:

DPODL=[∂ r∂ x

∂ r∂ y

∂ r∂ z

∂

∂ x∂

∂ y∂

∂ z]=[

cos sin 0

−sin θr

cosr

0 ] That is to say u an auxiliary field expressed according to the polar coordinates r , . The derivativeof u compared to the variables r , are analytically given. One writes them in matric form:

DUDPO=[∂ u1

∂r

∂u1

∂ ∂ u2

∂r

∂ u2

∂ ∂ u3

∂r

∂u3

∂ ]

Then, one writes the derivative in the local base e1 , e2 , e3 :

DUDL=[∂u1

∂ x

∂u1

∂ y

∂ u1

∂ z∂ u2

∂ x

∂u2

∂ y

∂u2

∂ z∂u3

∂ x

∂u3

∂ y

∂ u3

∂ z]=DUDPO .DPODL=[

∂ u1

∂ r

∂ u1

∂∂u2

∂ r

∂ u2

∂∂u3

∂ r

∂ u3

∂] .[ ∂ r∂ x ∂ r

∂ y∂ r∂ z

∂

∂ x∂

∂ y∂

∂ z]

It now remains to write the derivative of u in the total base E1 , E2 , E3 . For that, one notes Rthe matrix of the vectors of the total base written in the local base (matrix of passage):

Rm=e⋅Em

Are I and J total indices, j ème derived from i ème component of u is written then:

u i , j=∑=1

3

uR i , j=∑=1

3

u , j R iuR i , j =∑=1

3

∑β=1

3

u ,R j R iu R i , j In this expression, terms uα , β are the components of the matrix DUDL . The term Rαi , j appearsas the curve of the local base. It can be neglected if the curve of the bottom of crack is low. In practice,its taking into account almost does not change the results.

5.2.3 Discretizations




6bd3f8cf6f8b

Just as previously, one notes S the curvilinear X-coordinate in bottom of crack. The calculation of thelocal value of K I , K II and K III require to solve the following variational equation for several fields

i :

1−2

E∫

0K I s i s ⋅m s ds=g u ,u IS i ∀ i∈[1, P ]

1−2

E∫

0K II s i s ⋅m s ds=g u ,u IIS i ∀ i∈[1, P ]

12∫

0K III s i s ⋅m s ds=g u ,u IIIS θ i ∀ i∈[1, P ]

éq. 5.2.3-1

Scalar fields K I s , K II s and K III s are discretized on the noted basis p j s 1≤ j≤N :

K I s = ∑j=1

N

K I j p j s

K II s = ∑j=1

N

K II j p j s

K III s = ∑j=1

N

K III j p j s

For the fields theta, the approximation [éq. 5.1.3-1] is preserved.

Just as for G , by injecting the expressions of the discretizations in the variational equation [éq. 5.2.3-1], it comes the linear system:

∫ 0

∑j=1

N

K I j p j s∑k=1

M

ki qk s ⋅m s ds = g u ,uI

S i , ∀ i∈[1, P ]

∫0

∑j=1

N

K II j p j s ∑k=1

M

ki qk s ⋅m sds = g u ,u II

S i , ∀ i∈[1, P ]

∫0

∑j=1

N

K III j p j s∑k=1

M

ki qk s ⋅m sds = g u , uIII

S i , ∀ i∈[1, P ]

5.3 Method G-theta with X-FEM

Until now, the method G-theta was presented within the framework of the finite element method (MEF).To adapt the method G-theta to framework X-FEM, there are very few modifications to be made. Theonly difference resides at the level of digital integrations, to calculate the integrals of fields [éq. 5.1.2-2].Just as for the integration of the tangent matrices of second members, one carries out integration onthe subelements.




6bd3f8cf6f8b

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7 Description of the versions

Index document Version Aster Author (S) Organization (S) Description of the modificationsWith 9.4 S.GENIAUT, P.MASSIN

EDF/R & D AMAInitial text

B 11 D. COLOMBO (University ofManchester), Mr. GUITON, Mr.SIAVELIS (IFPen), S.GENIAUT, V.X. TRAN EDF/R& D AMA, P.MASSINEDF/R & D LaMSID

Initial text

C 11 S.GENIAUT, EDF/R & D AMA Separation of the parts on thecontact and the propagation

D 11 Mr. NDEFFO, P. MASSIN, EDFR & D LaMSID

Addition of the quadratic elements3D


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