Note of use of the boundary conditions treated by elimination

Transcription

1 Titre : Notice d'utilisation des conditions aux limites tr[...] Date : 01/10/2012 Page : 1/5 Note of use of the boundary conditions treated by elimination Summarized the processing of the boundary conditions of the Dirichlet type by elimination (AFFE_CHAR_CINE) does not offer the same generality as by dualisation (AFFE_CHAR_MECA for example). This processing is used when the goal is to improve the execution time of a computation, or if working with positive definite matrices is desired. Note that the boundary conditions available in AFFE_CHAR_* (* = MECA/THER/ACOU) cannot all be eliminated and treated by AFFE_CHAR_CINE. This document describes how to use the kinematical loads in the Aster command sets. There are 3 cases (from the simplest to the most intricate): a global command is used (THER_LINEAIRE, STAT_NON_LINE, ). In this case, kinematical loads are used like other loads. an eigen modes computation is performed. It is then necessary to add an argument in the command ASSE_MATRICE. a step by step computation is performed and the linear systems are solved with the commands FACTORISER and RESOUDRE. In the latter case, the command CALC_CHAR_CINE should be used.

2 Titre : Notice d'utilisation des conditions aux limites tr[...] Date : 01/10/2012 Page : 2/5 1 Principle of elimination One seeks to solve in R n the problem of minimization under stress (Pb1) according to: min u U G 1 2 ut K u u T f with U G = {u R n, u G = } where R p is known ( 1 p n ) G is the subset of N ={1,..., n}, of cardinal p : G=g 1... g p u G is the projection of u on under space generated by {u i } i G where u i j = ij, j N K is a symmetric matrix n n, f R n is built-in. The contraintereprésente u G = of the boundary conditions of the homogeneous Dirichlet type or not. If one notes L=C N G the complementary one to G in N, one can, using u i previously definite, to all in all break up R n direct V G = vector space generated by {u i } i G and of V L = vector space generated by {u i } i L ; Consequently, we have R n =V G V L and it is noted u=u G u K where u G = u G and = u L still in vectorial notation u= {u G } the problem is (Pb1) can thus be written in the form of the problem (Pb2): {min 1 u G V G V L u G = which is equivalent as: Pb1 Pb2 2 u T G K GG u G 1 2 u T L K LL u T L K LG u G u T L f L u T G f G {min 1 2 V L u= T K LL T K LG T f L One then eliminated u G from the problem of minimization.

3 Titre : Notice d'utilisation des conditions aux limites tr[...] Date : 01/10/2012 Page : 3/5 The matricial problem associated with (Pb3) follows : One searches minimizing 1 2 u T L K LL u T L K LG u T L f L what amounts solving the following matricial problems: K LL = F L K LG One can thus write: Pb1 Pb2 Pb3 [ K LL 0 0 I G][ u G] = [ f K LG 0 ], in K ' [ u G] = f ' 2 Aster the kinematical loads 2.1 a kinematical load (standard Aster is Processing : char_cine_* [* = meca/ther/acou]) makes it possible to characterize the group G d.o.f. imposed and for i i G which are the values assigned to these d.o.f. f. The definition of a kinematical load is done via the operator AFFE_CHAR_CINE for constant or function of geometry or time. (u_0) _i 2.2 The kinematical vectors the kinematical vector is a cham_no_* which represents the vector [ 0 ]. To each kinematical load corresponds a kinematical vector. This operation is carried out by the operator CALC_CHAR_CINE. 2.3 Computation of K' K' is directly calculated at the assembly time by the operator ASSE_MATRICE provided naturally whom one provides in argument a list of kinematical loads. The data structure MATR_ASSE_* has been modified in order to be able to store K' when it is necessary. 2.4 Computation of f' After L '' operator TO FACTORIZE the concept of the matr_asse_* type produced, contains factorized K' and unchanged matrix KLG. The computation of F'is carried out at the time of the resolution: it is necessary to provide the operator TO SOLVE in argument the kinematical vector corresponding to This operator calculates then f ' before solving fact K ' [ u G ] = f '. [ 0 ] via key word CHAM_CINE.

4 Titre : Notice d'utilisation des conditions aux limites tr[...] Date : 01/10/2012 Page : 4/5 3 Examples of command files 3.1 Mechanical computation with a global command (STAT_NON_LINE): DEPIMP=AFFE_CHAR_CINE (MODELE=MOD, MECA_IMPO=_F ( GROUP_MA = LCD1, DY = -2.0)) RESU=STAT_NON_LINE (MODELE=MOD, CHAM_MATER=CHMAT, EXCIT= _F ( CHARGE = DEPIMP, ) FONC_MULT = FONC), 3.2 Kinematical loads for a computation of eigen modes: CHARCINE=AFFE_CHAR_CINE (MODELE=MODEL, MECA_IMPO=_F (GROUP_MA=' GM2', DX=0.0, DY=0.0)) KASS=ASSE_MATRICE (MATR_ELEM=KELEM, NUME_DDL=NUMÉRIQUE, CHAR_CINE=CHARCINE,); MASS=ASSE_MATRICE (MATR_ELEM=MELEM, NUME_DDL=NUMÉRIQUE, CHAR_CINE=CHARCINE,); # computation of the eigen modes of structure MODES=MODE_ITER_SIMULT (MATR_RIGI=KASS, MATR_MASS=MASS, CALC_FREQ=_F ( NMAX_FREQ=10,)) 3.3 Computation step by step using commands FACTORISER and RESOUDRE : CHCINE=AFFE_CHAR_CINE ( MODELE=MO, MECA_IMPO= ( _F (GROUP_NO = SUPY, DY = 0.), _F (GROUP_NO = CHARGE, DX = -1.))) MEL=CALC_MATR_ELEM ( MODELE=MO, CHAM_MATER=CHMAT, OPTION=' RIGI_MECA') NU=NUMÉRIQUE_DDL ( MATR_RIGI=MEL) MATAS=ASSE_MATRICE (MATR_ELEM=MEL, NUME_DDL=NU, CHAR_CINE=CHCINE) SCMBRE=CRÉA_CHAMP ( ) VCINE=CALC_CHAR_CINE (NUME_DDL=NU, CHAR_CINE=CHCINE ) MATAS=FACTORISER (reuse=matas, MATR_ASSE=MATAS)

5 Titre : Notice d'utilisation des conditions aux limites tr[...] Date : 01/10/2012 Page : 5/5 DEP=RESOUDRE (MATR=MATAS, CHAM_NO=SCMBRE, CHAM_CINE=VCINE)