Code_Aster. Detection of the singularities and computation of a card of size of elements

Transcription

1 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : /6 Responsable : Josselin DLMAS Clé : R Révision : 9755 Detection of the singularities an computation of a car of size of elements Summarize: One proposes here a metho which aims at improving the processing of the singularities in the strategies of mesh aaptation with the software HOMARD (in the case of refinement) or with software GMSH (case of mening of meshes). This mechanism allows, on the one han to etect the finite elements connecte to singular zones an on the other han to obtain, for a given total error, the size of the finite elements of the new mesh in the event of mening of meshes. This functionality is accessible in comman CALC_RRUR by computation options SING_LM (constant fiel by element) or SING_LNO (fiel at noes by element). This option is vali only in mechanics. It is necessary to have calculate beforehan an estimator of error in mechanics an strain energy on each element. In any rigor, this metho is vali only in elasticity.

2 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : /6 Responsable : Josselin DLMAS Clé : R Révision : 9755 Contents Introuction... 3 Detection of singularities Principle of metho Detection of the singular noes valuating about the singularity Cases of imension Cases of imension xtension to the zones of stress concentration Construction of an optimal mesh General information Definition of optimality Determination of an optimal mesh classical stimators of error Cases of the regular solution Cases of the singular zones stimators of error in quantities of interest Cases of the regular solution Cases of the singular zones Use into Coe_Aster commans Perimeter of use Bibliography Description of the versions of the ocument... 6

3 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : 3/6 Responsable : Josselin DLMAS Clé : R Révision : 9755 Introuction the purpose option suggeste here is to improve the processing of the singularities in the strategies of mesh aaptation suggeste in the Coe_Aster. Inee, the presence of singularities (present in practice in any real structural analysis by finite elements) implies two kins of ifficulties which one will escribe here as theoretical an of practices. The theoretical ifficulties come owing to the fact that the contribution to the error in energy of the elements touching a singularity is form C h ( C a constant, h size of the element an the orer of the singularity) while the contribution of the elements except singularity is form C h q ( q epening only on the egree of interpolation of the shape functions of the element). The aaptation of the mesh must take into account this ifference to be most effective possible. For example, to ivie the contribution of the error by 4, it will be necessary in the case of to take elements 6 times smaller a crack ( =/ ) an elements times smaller in the case except crack with quaratic elements ( p= ). The practical ifficulties come owing to the fact that, in zones of singularities, the contributions to the error in energy are important. If one aims at obtaining an error in weak energy, these zones thus shoul be very strongly refine. However, one can woner about the influence of these errors in energy on physical quantities which interest the engineer (isplacement in such point, maximum stress in such sensitive area, etc ). In other wors, it is not because the zones of singularities cause important errors on the energy which they have a great influence on computation apart from these zones. In practice, the estimators of error quickly inicate the only zones of singularities as being refining: the zones of singularities mask the other errors, for example a zone with strong graient which one woul wish to refine. The Laboratory of Mechanics an CAO of Saint-Quentin evelope a metho making it possible, on the one han, to etect the singularities, an on the other han, to etermine, for a given total error, the size of the finite elements of the new mesh in the event of mening of meshes. The use of this two information can be uner consieration uner two angles: The finite elements consiere as singular by the metho can be exclue from the process of cutting, the new size of the finite elements is given to a r lor so that this one buils the new mesh by as well as possible respecting this new car of size. Currently, the software HOMARD cuts out the element once (for example into D, a triangle is ivie into 4 but not more). To continue cutting, it is necessary to call on HOMARD again. An evolution is thus to envisage so that one can ivie several times an element an thus as well as possible respect the car of size of the new mesh. It is however possible to use the free mesh generator GMSH which irectly takes a car of size in entry. Note: This ocument shows for the group the note resulting from a CRCO between the LMCAO an the epartment AMA whose reference is quote in bibliography ([bib]).

4 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : 4/6 Responsable : Josselin DLMAS Clé : R Révision : 9755 Detection of the singularities. Principle of the metho When the exact solution of the stuie problem present of the singularities, the orer of convergence of the solution finite elements is moifie an thus also that of the estimator of error. Let us consier, for example, a problem of plane elasticity iscretize with triangular elements of egree p. If the exact solution U ex is regular, it is known that ([bib], [bib3]): u u h = e C h p éq.- Where e C h p is the contribution to the error in energy, is: e e h K e h éq.- On the other han, if the exact solution presents a singularity, for example if, locally in the vicinity of a point M 0, the fiel of isplacement is form (with r an polar coorinates in the vicinity of the point M 0 ): Then, one shows that [Strang & Fix, 976]: U ex =r V W avec 0 éq.-3 e h C h éq.-4 It results from it that the rate of convergence of the total error in energy becomes inepenent of the egree p of the finite elements use an it is the same of that of the measurement of the error (for example, so p=ou then =/ for a crack). In orer to obtain a goo preiction of the optimize meshes, the preceing observations lea us to use a rate of convergence q per element such as the estimator of error checks: =O h q éq.-5 a way simple to efine these local coefficients consists in taking: q = if the element is connecte to a singularity of orer ; q =q for all the other finite elements where q epens only on the type of finite elements use. The metho presente thereafter thus comprises three phases: etection of the singular zones, in fact singular noes of the mesh; numerical evaluating of the coefficient q for the elements connecte to the noes consiere as singular (for the other elements, one fixes then q =q ); computation of the coefficient of moification of size r.. Detection of the singular noes the iea is to use the site errors. Inee, the experiments show that these site errors present a peak in the vicinity of a singularity. For each noe i of the mesh, one thus compares the average error m i

5 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : 5/6 Responsable : Josselin DLMAS Clé : R Révision : 9755 of the elements connecte to the noe i with the average error M on the group of structure. The noe i is regare as singular if: m i M éq..- with m i = connecté à i connecté à i mes an M = structure structure mes éq..- where is a coefficient larger than an mes surface it in D or volume in 3D of the element. The numerical experiments showe that the singular noes are well etecte while fixing = in imension, =3 in imension 3 for of the finite elements linear an = in imension 3 for of the finite elements quaratic. Notice : Notice : From a numerical point of view, the etection of the singular noes iffers between the cases D an 3D. The conitions given thereafter for this etection are not base on a particular theory but rather on the experience gaine in this fiel by the Laboratory of Mechanics an CAO of Saint-Quentin. In D: a noe I am regare as singular if he meets the 3 following conitions: m i M m i i m éq..-3 m i 3Min m i, m 3 i Where m i, m i an m 3 i is the averages of the error for the elements belonging to layers, an 3, respectively, compare to the noe i consiere. The layers are efine as follows: Lay own : elements which have noe I to test, Layer : elements in contact (face, ege or noe) with an element of layer, Layer 3: elements in contact (face, ege or noe) with an element of layer. In 3D: the noe i is regare as singular if he meets the conition m i M an if one of the noes connecte to the noe i consiere meets the conition m i Noeu connecté à i M. Contrary to the case D, the noe i is singular only if one of its neighbors is also (one forgets the singular noes isolate to keep only the singular eges). In D, only the noes tops are examine. In 3D, only the noes tops locate on an ege of structure are examine (only for reasons of time computation; this conition coul thus be moifie)..3 valuating about the singularity For each etecte singular i noe, the orer of the singularity, i.e. the value of q which will be use for the elements connecte to the noe i, is given by ientifying the value of the ensity of energy of the solution finite elements in the vicinity of the noe i with the theoretical value in the vicinity of a singular point..3. Case of imension

6 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : 6/6 Responsable : Josselin DLMAS Clé : R Révision : 9755 In this case, one calculates energy finite elements average, in iscs A of center i an r : e h r = mes A A u h K u h A éq.3.- While ientifying, by a metho of least squares, this average energy with the theoretical value in the vicinity of a singularity of orer : one numerically obtains a value close to. e r =k r c éq.3.- e h r 0 A r/r 0 0 Appear.3.-a: Numerical evaluating of In practice the numerical experiments show that it is enough to carry out the ientification in a zone corresponing to 3 layers of elements aroun the singular point an to evaluate e h r for 5 to 8 values of r regularly istribute in this zone (we took 0 values)..3. Case of imension 3 In 3D the situation is more complex. The point singular, generally, are not isolate an it is thus frequent to be in the presence of singular eges. Let us consier, for example, the case of a cube embee on a face an subjecte to tractive efforts: all the points of the eges of the clampe face are singular [Figure.3.-b].

7 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : 7/6 Responsable : Josselin DLMAS Clé : R Révision : 9755 Appear.3.-b: Cubic embee in tension In this situation, the evaluating of average energy in balls A of raius growing an centere on a singular noe oes not make it possible to ientify q. Inee, as the raius increases the extent of the singular zone containe in the ball A increases an one oes not obtain a fast ecrease of e h [Figure.3.-c]. zone à fort graient boule A Appear.3.-c: nergy in concentric balls When the singular points are not isolate, it is necessary to ientify the coefficient q by calculating the ensity of energy in coaxial cyliners built on the eges whose ens were regare as singular [Figure.3.-] an cf notices [.]. Appear.3.-: nergy in coaxial cyliners.4 xtension to the zones of stress concentration In practice, we note that the preceing metho, clarification on cases presenting of the singularities also makes it possible to take into account correctly the zones with strong graients of stresses even if mathematically these zones o not correspon to singularities.

8 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : 8/6 Responsable : Josselin DLMAS Clé : R Révision : Construction of an optimal mesh 3. General information the purpose of a proceure of aaptation is to guarantee to the user a level of accuracy on the total error while minimizing the costs of computation. To evaluate the errors of iscretization, one an the uses a relative total measurement of the error local contributions associate with: = éq 3.- the iea is an to use the results of this first analysis finite elements estimators of errors to etermine an optimal mesh T i.e. a mesh which makes it possible to respect esire accuracy the while minimizing the costs of computation. One buils then the mesh T using an automatic mesh generator an one carries out one secon analysis finite elements. 3. Definition of optimality For a given total error 0, a mesh T is optimal compare to a measurement of error if: = 0 précision emanée N nombre 'éléments e T est minimum éq 3.- This criterion of optimization naturally results in minimizing the costs of computation. 3.3 Determination of an optimal mesh to etermine the characteristics of the optimal mesh T, the metho consists in calculating on each element of the mesh T a coefficient of moification of sizes: Where h is the current size of the element an h force on the elements of T r = h éq 3.3- h the size (unknown) which it is necessary to in the zone to ensure optimality [Figure 3.3-a]. A possible choice to efine the size of an element h is to take the size of largest with imensions of this element. The etermination of the optimal mesh is thus brought back to the etermination, on the initial mesh T, of a car of coefficients of moification of size.

9 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : 9/6 Responsable : Josselin DLMAS Clé : R Révision : 9755 T h h* * T * Appear 3.3-a: Definition of the sizes The computation of the coefficients r is base on the rate of convergence of the error: =O h q éq 3.3- where q epens on the type of finite element use but also on the regularity of the exact solution of with the ealt problem. For the classical estimators of error, one supposes that the rate of convergence of the estimator of error is equal to the orer of convergence of the solution finite elements. For the estimators in quantity of interest, this rate of convergence is equal to the ouble about convergence of the solution finite elements ([bib4]). Thereafter, to compute: the coefficient of moification of size r, one istinguishes the case from the regular solution ( q epens only on p, egree of interpolation of the shape functions of the element) of the case of the singular solution ( q epens only on, orer of the singularity of the fiel of isplacement). 3.4 Classical estimators of error One esignates by classical estimators of error the estimators who provie a norm (norm - L, norm - H, norm in energy) of the error in solution Case of the regular solution Initially, we suppose that the exact solution is sufficiently regular so that the value of q epens only on the type of finite elements use an is equal to the egree of interpolation use p ( p is worth for of the finite elements linear an for of the finite elements quaratic). In this case, to preict the optimal sizes, it is written that the ratio of the sizes is relate to the ratio of the contributions to the error by: =[ h ] h =r p éq where represents the contribution of the elements of T locate in the zone, i.e.: = [ ε ] / éq is the error of the element calculate on the mesh T.

10 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : 0/6 Responsable : Josselin DLMAS Clé : R Révision : 9755 The square of the error on the mesh T can thus be evaluate by: = r p éq an the number of elements from T : N = r éq Where is the imension of space (in practice, = or 3 ). Inee, r = h = h V / N N / V with N the number of element of T in (thus ), N the number of element of T in the zone of. One thus has N = r, that is to say N = N = The problem to be solve is thus: r the nombre total of elements of T. Minimiser N = r avec r p = 0 éq It acts of a problem of optimization with a stress on the variables of optimization. Introucing a multiplier of Lagrange, note A, the problem [éq ] reverts returning extremum the Lagrangian one: L {r } T conitions of extremality give: ; A = r A r p 0 éq the L = r r A p r p =0 T éq From where: r = [ ]/ p A p éq While eferring in the secon equation of [éq ], one from of euce A : A= p[ / p ] p / p 0 éq One replaces the statement of A thus obtaine in the equation [éq ] to obtain r :

11 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : /6 Responsable : Josselin DLMAS Clé : R Révision : 9755 r = / p [ 0 / p / p ] / p éq Cases of the singular zones to preict the optimal sizes, one uses a rate of convergence q efine by element: =[ h ]q q =r h éq where represents the contribution of the elements of T locate in the zone, i.e.: = [ ] / éq the square of the error on the mesh T can thus be evaluate by: q = r éq an the number of elements of T is always evaluate by: the new problem to be solve is thus: N = r éq Minimiser N = r avec r q = 0 éq which is a problem of optimization with a stress on the variables of optimization. Introucing a multiplier of Lagrange, note A, the problem reverts returning extremum the Lagrangian one: L {r } T ; A = r A r q 0 éq the conitions of extremality give: L = r r A q q r =0 T éq From where: r =[ / q Aq ] éq

12 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : /6 Responsable : Josselin DLMAS Clé : R Révision : 9755 While eferring in the secon equation of [éq ], one obtains a nonlinear equation in A (because q epens on the elements): [ [ q / q A q ] ] / q 0=0 éq It is solve by the metho of Newton (the multiplier of Lagrange is initialize by taking the multiplier of Lagrange of the regular solution i.e. the statement [éq ] with q = p ). Once A calculate, one from of euce r by the equation [éq ]. 3.5 stimators of error in quantities of interest One esignates by estimators of error in quantities of interest the estimators who provie an error on a precise physical quantity (quantity of interest) on a selecte zone Case of the regular solution In the case of the estimators in quantity of interest, the value of q is worth p [bib4] ( p is worth for of the finite elements linear an for of the finite elements quaratic). To preict the optimal sizes, it is written that the ratio of the sizes is relate to the ratio of the contributions to the error by: =[ ε ε h ] p p =r h éq where ε represents the contribution of the elements of T locate in the zone, i.e.: is the error of the element calculate on the mesh T. ε = ε éq The error on the mesh T can thus be evaluate by: an the number of elements from T : = r p éq N = Where is the imension of space (in practice, = or 3 ). The problem to be solve is thus: r éq Minimiser N = r avec r p = 0 éq There still, it acts of a problem of optimization with a stress on the variables of optimization.

13 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : 3/6 Responsable : Josselin DLMAS Clé : R Révision : 9755 Introucing a multiplier of Lagrange, note A, the problem [éq ] reverts returning extremum the Lagrangian one: L { r } T ;A = conitions of extremality give: r A r p 0 éq the L = r r A p r p =0 T éq From where: r =[ / p A p ] éq While eferring in the secon equation of [éq ], one from of euce A : A= p[ / p ] p / p 0 éq One replaces the statement of A thus obtaine in the equation [éq ] to obtain r : r = / p [ 0 / p / p ] / p éq Cases of the singular zones to preict the optimal sizes, one imposes now: =[ h ]q q =r h éq where represents the contribution of the elements of T locate in the zone, i.e.: = éq the square of the error on the mesh T can thus be evaluate by: q = r éq an the number of elements of T is always evaluate by:

14 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : 4/6 Responsable : Josselin DLMAS Clé : R Révision : 9755 the new problem to be solve is thus: N = r éq Minimiser N = r avec r q = 0 éq which is a problem of optimization with a stress on the variables of optimization. Introucing a multiplier of Lagrange, note A, the problem reverts returning extremum the Lagrangian one: L {r } T ; A = r A r q 0 éq the conitions of extremality give: L = r r A q r q =0 T éq From where: r =[ / q Aq ] éq While eferring in the secon equation of [éq ], one obtains a nonlinear equation in A : 0 =0 éq [ [ q / q A q ] ] / q It is solve by the metho of Newton (the multiplier of Lagrange is initialize by taking the multiplier of Lagrange of the regular solution i.e. the statement [éq ] with q = p ). Once A calculate, one from of euce r by the equation [éq ]. 4 Use in Coe_Aster 4. the commans the orer of the singularity an the car of moification of size are calculate by the comman CALC_RRUR by activating options SING_LM (constant fiel by element) or SING_LNO (fiel at noes by element). Option SING_LM calculates, on each element, two components: DGR : the orer of the singularity i.e. the value of the coefficient q (which is worth p if the element is not connecte to a singular noe an who is worth if not); RATIO : the relationship between the current size h an the new size h of the finite element ( h /h =/r ); TAILL : new size h of the finite element ( h =r h ).

15 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : 5/6 Responsable : Josselin DLMAS Clé : R Révision : 9755 The computation of this option requires, as a preliminary, the computation of an estimator of error (it is the absolute component which is use an it is coe into tough in Coe_Aster) an of total strain energy. If one of these options is not calculate, an alarm message is transmitte an option SING_LM is not calculate. The user can inform optional key wor TYP_STI by inicating one of the options following: RM_LM for the estimator base on the resiues; RZ ( or ) _LM_SIGM for the estimator base on the smoothe stresses (Zhu - Zienkiewicz version or ); QIR_LM for the estimator in quantity of interest base on the resiues; QIZ ( or ) _LM_SIGM for the estimator in quantity of interest base on the smoothe stresses (Zhu - Zienkiewicz version or ). If this key wor is not then inicate the estimator base on the resiues RM_LM is chosen by (transmitte alarm message). If the two estimators of Zhu-Zienkiewicz are present, one chooses RZ_LM. For total strain energy, one uses: With STAT_NON_LIN: TOT_LM which is total strain energy on a finite element (vali for an elastic behavior an an elastoplastic behavior VMIS_ISOT_XXX ). With MCA_STATIQU : POT_LM which is the potential energy of elastic strain on a finite element an integrate starting from isplacements an temperature (vali only for one elastic behavior). The user must also inform the key wor PRC_RR which makes it possible to calculate the accuracy 0 of the equation [éq.-] in the following way: 0 =PRC_RR rreur totale. The value of PRC_RR strictly lies between 0 an (a fatal message is transmitte if this conition is not checke). For the option SING_LNO, it acts of a recopy of the values of SING_LM to the noes of the element. The computation preliminary of SING_LM is thus necessary. If SING_LM is absent, an alarm message is transmitte an option SING_LNO is not calculate. 4. Perimeter of use the perimeter of use is the same one (but more reuce) than that of the estimator of error chosen namely: For the estimator in resiue: finite elements of the continuums in D (triangles an quarangles) or 3D (only tetraherons) for an elastoplastic behavior, the estimator of Zhu-Zienkiewicz: finite elements of the continuums in D (triangles an quarangles) for an elastic behavior. In any rigor (cf [ ]), computation about the singularity is obtaine from theoretical energy at a peak of crack [éq.3.-], vali equation only in elasticity. The use of this option in elastoplasticity is thus to hanle with pruence.

16 Titre : Détection es singularités et calcul une carte [...] Date : 0/0/0 Page : 6/6 Responsable : Josselin DLMAS Clé : R Révision : Bibliography [] COORVITS P.: Mechanism of etection of the singularities. First part. Note Laboratory of Mechanics an CAO (Saint-Quentin). [] CIARLT P. - G.: The finite element metho for elliptic problems, North-Hollan, 978. [3] STRANG & FIX: Year analysis of the finite element metho, Prentice hall, 976. [4] stimators of error in quantities of interest. [R4.0.06] 6 Description of the versions of the ocument Inex Doc. Aster Author (S) or contributor (S), organization Description of the moifications A 8.4 V.Cano DF/R & D /AMA initial Text B 9.4 J.Delmas DF/R & D /AMA Recasting of the ocument + aition of the estimators of error in quantities of interest