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P2$ !TҮpU̓5da9P2$D-N]#Uqø9P2$!u?REq;)=P2$s(>}fP2$D\jzýW|P2$S_6Ѡv nP2$H{?[ AB3^P2$[NY7ЖQP2$V:fG/fP2$-Շ4RBiP2$lJDýuP2$go-)e piyP2$޶kdFzyPc $`33@337 )F ʚ;2Nʚ;g4:d:dv 0bppp0 <4!d!d` 0b5b<4dddd` 0b5b? %;  Outline  tState of the art in symbolic finite elements computation Semi-analytical/numerical approaches Enhancing finite element code performances Speeding up finite element code development Symbolic approaches and Object-Oriented programming for Finite Elements Hybrid Symbolic/Numerical Approach for Finite Elements: basic principles The FEM_Theory and FEM_Object approach The approach An example: A space  time formulation for the incompressible Navier-Stokes problem Constitutive law Modeling The object  Materiau Symbolic approach for constitutive law modeling Conclusions9|Hq`F 9|Hq ` F         r7>$Semi-analytical/numerical approachesB      Semi-analytical/numerical approaches Programming of the FEM in a symbolic computation environment: Maple, Mathematica& Enables us to keep symbolic values Example: Ioakimidis 1993 A parametric study in linear elasticity Principle: square domain, symmetric loads, 8 triangular elements Expansion in Taylor series Resolution of the linear system using Gauss-Seidel Method Solution in symbolic form with variable iteration numbers Rem: Completely symbolic 4 dofs !!!j%u%u  bj - o)Enhancing finite element code performance(   MEnhancing finite element code performance utilisation of a classical symbolic environment to enhance expressions by simplifying computations Example: Yang 1994 (see also Yagawa 1990) Numerical integration scheme Principle: Gauss quadrature: If f polynomial of degree 5: exact integration for n=3 48 multiplications and 17 additions Effectuate all algebraic operations: dependance on Gauss coordinates and material caracteristics 6 multiplications and 5 additions Extensions to fractions and to 2D computations Rem: Enhancement of the performances but completely problem dependent L*ZeZ*ZZ ZLZ$ZcZ#Z/ZZCZ*e*     .      $  c#/C~* w4  "SB+Speeding up finite element code developmentP       Automation of the development of FE models: automatic generation of code to compute elemental contributions Example: Noor 1981 (see also Wang 1981) Development of stiffness contributions for a thick shell element  Total Langrangian formulation  Displacement and mixed formulations (Hellinger-Reissner) Principle: Displacement formulation  Total potential energy: With: Finite element model Program in MACSYMA: entering numerical values: number of dofs, shape functions FORTRAN code for evaluation of the different finite element contributions Rem: generation for given elemental contributions, treatment of a new problem3 *N3 * Nwc 3 FQ]KGSymbolic approaches and Finite Elements Codes Advantages and Drawbacks .(1) S.A./N. Complete development of the FE model in a single and homogeneous environment Parametric studies: influence of parameters (geometric, materials& ) Applicable to small problems (less than 100 Dofs) Extension to NL problems difficult (2) Enhancement of FE codes performances Usefulness of algebraic computation systems to optimize the FE codes (in term of speed) Possibility to automatically generate FE code ( bug free !!) Only local action on the FE code (3) Speeding up FE code development Time spent to develop FE code drastically decreased Automatic production of FE code: Lack of flexibility, difficult to integrate new FE formulation Heavy systems in general (see FINGER  P.S. Wang 1981): multiple systems, and langages For a new problem, all the process is to be repeated ZZ)ZZ$Z Z V)"$U  ,  5 IHybrid Symbolic/Numerical Approach for Finite Elements: basic principlesj         High potential of symbolic approaches Taking advantages: wide range of problems, systematic approach for FE developments Avoid drawbacks: size of problems and extension to NL problems Idea: integrate the strong, weak and discrete forms of the FEM into a computerized environment AND use a classical FE code Power of symbolic concepts Efficiency of numerical codes Integration in natural way for the numerician (mechanics, mathematics& ) Problem: How to easy integrate these concepts in a computer? Integration in an OO environment Structuring of data simplified Hierarchy: codes more compact Natural way of programming&}=!X&}=! X  NHybrid Symbolic/Numerical Approach for Finite Elements: FEM_Theory/FEM_Object~           A Space-Time Formulation For Incompressible Navier-Stokes Formulation  Approximated Form+( *A numerical example: dam break problem (1)N        *A numerical example: dam break problem (2)N         Conclusions   fSur les dveloppements prsents& Seuls les aspects lis aux formulations E.F ont t abords Aspects constitutifs, aspects algorithmiques de base Efficacit des codes E.F. avec ces approches Problme du choix TECHNIQUE des outils utiliss (SMALLTALK/C++ ; Gibiane/FORTRAN-ESOPE). Les ingrdients pour aller plus loin& Concepts Intgrer au maximum les dveloppements dans un environnement informatis afin d viter une rupture dans les dveloppements d outils de simulation Technique Environnement homogne tant au niveau conceptuel que programmation Une seule approche informatique: paradigme objet Portabilit et distribution aise: LE WEB Efficacit du code acceptable & et JAVA!!!!!"Z<Z5ZZ&Z ZZ ZCZ1ZIZZ"<5& C1I&    Conclusions   Down: Only aspects linked to the formulations taken into account Constitutive aspects, algorithms,& Problems linked to the crucial choice of the tools (portability& ) Efficiency of the code generated Up: Homogeneous environment A single computerized approach: Object-Oriented paradigm Easy extendability to new formulations No preliminary hand studyv;#d;#d# 0'Interest of an Object-Oriented ParadigmJObject-Oriented programming Hierarchical organization of classes with inheritance Encapsulation of data and behavior inside the Object Communication between objects by sending messages (polymorphism) Application of the O.O.P. to the FEM Enhancement of the modularity of FE codes Better maintenability of FE codes Better extendability of FE codes Natural representation of algorithms and easy manipulation of elemental contibutions BASIC IDEA Today, one can integrate in existing finite element codes symbolic manipulations concepts linked to special problematic aspects of the FEMZZ%Z*ZCZYZ ZZ%*CY H T  A Space-Time Formulation For Incompressible Navier-Stokes Formulation  Strong FormB  A      7Finite Element code generated for Navier-Stokes STF (1)! 7Finite Element code generated for Navier-Stokes STF (2)B       /  ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@z?" dd@  " @ ` n?" dd@   @@``PR    @ ` ` p>>   k(    6Ǹ   e1Cliquez pour modifier le style du titre du masque2 2:  0ɸ P  vCliquez pour modifier les styles du texte du masque Deuxime niveau Troisime niveau Quatrime niveau Cinquime niveau4 w  0θ `  >*  0dԸ `T    @*  08ٸ `x  @*f  0"@P ZB  c $D"ppH  0@޽h ? ̙33 "Modle par dfaut0 xB(  x x 0 b P   j T*   x 0b    j V*  d x c $ ?  j> x 0j p @  j vCliquez pour modifier les styles du texte du masque Deuxime niveau Troisime niveau Quatrime niveau Cinquime niveau4w  x 6dj P  j T*   x 6cj   j V*  H x 0޽h.^ ? ̙33( `(  ` ` 0uj P2  j R*    ` 0yj  2 j T*    ` 6}j P  j R*    ` 60j   j T*   H ` 0޽h.^ ? ̙33   D < 0- (  R   s *x  c $b  b   C tA\C:\Mes Documents\Documents\ISTIL\Logoisti.jpgP    C |AdC:\Mes Documents\Documents\ISTIL\WEB MCS\ucbl.gif@   0b  QEnvironnement symbolique de dveloppement en mcanique, Orient objet et portableRR, R   <C:\Mes Documents\cdcspLogo.gif ?H  0@޽h ? ̙33   $(  r  S b  b r  S db0 b H  0@޽h ? ̙33L  (  r  S Tj  j r  S jP j `  c $A ??@  x  <A ??  x  <A ??   x  <A  ??/ <   x  <A ??   H  0@޽h ? ̙33L   (   r  S Бj  j r  S jp j `   c $A ???@  x   <A +??  +x   <A %??8 <0  %H  0@޽h ? ̙33  |t$ (  $r $ S Зj  j r $ S jP j ` $ c $A .??pP  .x $ <A ???j L  ?H $ 0@޽h ? ̙33  ($(  (r ( S ,j  j r ( S jP j H ( 0@޽h ? ̙33  0$(  0r 0 S جj  j r 0 S jP j H 0 0@޽h ? ̙33|   , $ 4 (  4r 4 S ab  b  4 0.b`R ` Model 1 Differential system Lu=f$! 4 02b  &Model 2 Variational form a(u,v)+b(v)=0$'  4 0AbP B  |Model 3 Discrete form Kd=f$ 4 <H:b  [Model 4 Finite Element code$^ 4 6>0 ^  4 633> PS  4 <XDb  } FEM_Theory Expression Term Integral IntEquation DiscretizedExpression DiscretisationMatrix DiscretizedEquation4~o)U  4 <ob 0 ~*Automatic generation of code FEM_Object8+ 3333RB  4 s *D>  `  4 c $A @??0 @x 4 <A A??P@L A` 4 c $A K??<Ll K` 4 c $A M??8 Mx 4 <A P?? p  Px 4 <A S??h   Sx 4 <A V?? (  VLB 4 c $D LB 4 c $D LB 4 c $D H 4 0@޽h ? ̙33  !g@O(  @K F 0\  G@   rr H@ BGc 1? &rB I@ B1? rB J@ B1? `  K@ Tjgֳgֳ? h  Kx  rr L@ BG1?nv lr M@ <G1??#6 k~B N@ N1?  ~B O@ N1?oo~B P@ N1? $ Q@ NA ??0 iX $0!0 R@ NA ??lX $0!0 S@ NA ??\X $0!0 T@ NA ??P^ X $0!0 U@ NA! ??  'X $0!0 V@ NA" ?? " pX $0!0 W@ NA" ??JX $0!0 X@ NA# ??jX X $0!0 Y@  fA% ? 1? U.  [@  fA' ? 1? pG  \@ HA( ??9nX $0!0B ]@  `D1?@@ ^@ Z8j 1?@ {Shakib 1987 Tezduyar 1992 $ B _@  `D1?@@B `@  `D1?  B a@  `D1?U U B b@  fD1?` l` B c@  `Df1? M   d@ c j0e0e 0  j  e@  fA, ? 1?P 0  f@ Z j 1?\n 0 g(Tezduyar 1992)  g@  fA& ? 1?0  H @ 0@޽h ? ̙33  h`  P(  Pr P S ,j  j  P  fA- ? 1?.   P  fA. ? 1? 7p NF d  P tA  x P H& SNA/?@d x P H& U9A0?   P Zj 1? @t=0.5s   P <8Zbpu t Dam break problem&      P <Tjp U v Numerical results(   H P 0@޽h ? ̙33  A90T(  Tr T S  j  j NF   T x T H>B(A1?  x T H7;H2A2? o T Z`j 1?7  ?t=3 s H T 0@޽h ? ̙33 G?@"#h(  h h T9e1?  O Cadre gnral du projet : projet dbut au CEA Saclay dans le code dans CAST3Mj q5 -4 P hh88 h 6>e&o dRelation de comportement   6R * , h3  f h 633* f h 633* ,~ h BA3 /??A2  /  h BIeOr  dDescription des variables:    ^  h 6. ^  h 6  h BPMeM  YModle matriau   ~  h BA4 0??@ 0 h B Qe 3 W Intgration :   6R @ 0 h3  L(f h 6@ 0f h 6@ 0 ~ h BA5 1?? @M 1 h BVe U  XCode de calcul   ~ h BA6 2??  2~ h BA7 3??6 ,  3^ h 6  ^ h 6 *^ h 6i$ vR h@ NZGHI. 4LUvR h NZG:HhI:. 4j~ h BA8 4??Cz 4^ h 633A ^ h 633E ~ h BA* ??D  5~ h BA: 6??$ : 6~  h BA; 7?? C : 7* !h TProcessus itratif de rsolution des quations de la mcanique? ? ? hh88 "h S 4gep  e q;Implantation Interactive de Lois Constitutives en Mcanique;   #h 64je00P /o Problmatique : - exemple: modle de mcanique des solides: plasticit crouissage cinmatique et isotrope -2 q2] p2] p H h 0@޽h ?/@ hhh hhh a( P l((  l l <we @  %* * -INC SMMODMAT * * OBJET DE TYPE "MODELE MATERIAU" * * * SEGMENT,MMATER CHARACTER*8 VAR_PRIM(NVARP,2) CHARACTER*8 VAR_DUAL(NVARD,2) CHARACTER*8 VAR_INT(NVARI,2) CHARACTER*16 FICH_MAT CHARACTER*4 CARAC_MAT(NCAR) ENDSEGMENT * 6%  % &  l HA< 8???J b  8 l HHve?c  cDescription des variables   d l <?A d l <?+A   l S e  e DObjet matriau  H l 0@޽h ? a( `!p(  p~ p BA4 9??r r+ 9 p B??C   > p Be`   dDescription des variables:    ^ p 6BB]^ p 6BB p Bek  [Modle matriau   ^ p 6 t ^ p 6I x  p 0e  BSI SQRT(1.5*dev(SIG) : dev(SIG)) - SIGS>=0 dSIG=(E/(1+NU)) ( dev(EPS) + (dev(SIG):dev(EPS)) / (dev(SIG):dev(SIG))) ) + E/(3(1-2NU)) tr(EPS) I FINSI SI dev(SIG):SIG<=0 dSIG=(E*dev(EPS))/(1+NU) + E/(3(1-2NU)) tr(EPS) I FINSI *                   ,       - C p <te8  q Exemple : plasticit parfaite- critre de Mises (dformations planes) Le principe : 61 -  r r x p <A= ???0 ?jb p BG*HI B X  p 0 B !p S e  e _)Approche symbolique pour loi constitutive)  H p 0@޽h ?/@ppppp p a(  @0(  x  c $j  j x  c $@jP j H  0@޽h ? ̙33  X$(  Xr X S |2j  j r X S 83jP j H X 0@޽h ? ̙33  P](  x  c $   x  c $d P  R  s *   < d  _-Application to abstract mathematical concepts..H  0@޽h ? ̙33  `8\(  8r 8 S     8 T@'gֳgֳ?fB ` v& Incompressible Navier-Stokes Problem:' ''  8  fA [? 1?@ e  [  8 NXbgֳgֳ? 0 *  Formulation: Galerkin space/time formulation, continuous in space, discontinuous in time, stabilized by adding least-squares type terms2 { { < KKhh  8 0+ ` `j C Remark: This formulation enables us to deal with moving boundaries2  ;  ;<  hhV8 $  8P l 2  8 Zq'1?D`   8  0e0e    B8CDE,F4 1 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 6`n%R308@P$k  8  fA \? 1? \ 8  fA ]? 1?$ ]B 8  fD1?`4B 8B  fD1?@pB 8  fD1?t0  8  fA ^? 1?t  ^ 8 # lA _? 1?pg 4  _B 8  fD1?m T- B 8  fD1? A B 8  fD1? S B 8  fD1?n .l B 8B  fD1?`t 8  fA `? 1?O `B 8  fD1? o B 8B  fDf1?TpD  8  fA a? f1?Hc aH 8 0@޽h ? ̙33H  pH(  Hr H S 2   y H <b0  FEMObject Dictionary Dof Domain FEMComponent Element PlainStrain STF_ELEMENT NewElement Load Node TimeItegrationScheme TimeStep LinearSystem Tools GaussPoint NewElementGaussPoint  )  V       "' H T@gֳgֳ?`! S FEM_Object : partial view of the hierarchy of classes of the numerical environmentT TT H H TE 1? p  Computation of the elemental contributions Computation and management of the elemental contributions at the level of the integration point  H NA) ??dX $0!0H H 0@޽h ? ̙333  Ls(  Lr L S (L    L T\Mgֳgֳ?< w h Class NewElementGaussPoint   & L T Ogֳgֳ? 9  VAttributes : - GradientU - VectorU Methods : - giveGradientU( ) - giveVectorU( )W WW  L TQgֳgֳ?* _ Class NewElement   L TXgֳgֳ? P  0computeGauss2Pts giveGauss2PtsArray ... computeTangentStiffnessMatrixAt( ) computeTangentStiffnessMatrixAt11 ( ) computeTangentStiffnessMatrixAt111 ( ) ... computeStiffnessMatrixAt ( ) computeStiffnessMatrixAt11 ( ) computeStiffnessMatrixAt111 ( ) computeStiffnessMatrixAt111FunctionAt ( ) & ComputeJumpTermAt() & giveJacobianMatrixAt ( ) giveJacobianMatrixAt11 ( ) ...s ss H L 0@޽h ? ̙33! T!L!p55t(  tQ t T|e1?.> 1 Description symbolique pour la loi constitutive N1 q -0 2 hh88: t  fȿe..1))? <Implantation Interactive de Lois Constitutives en Mcanique = = = ..]] t 0Le p HStructures pour le calcul symbolique Relation de comportement: ensemble de couples condition/quations (systme d quations) Equation : membre de droite et membre de gauche Expression : description arborescente base sur les oprateurs et les variables deux objets permettent de dcrire la structure : segment oprateur segment variable % /  =    &   { $    %l&  {$&   ng`f t s *A> @??   @X t 0 R@ L ^#  t# pt t 0A? 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P2$ !TҮpU̓5da9P2$D-N]#Uqø9P2$!u?REq;)=P2$s(>}fP2$D\jzýW|P2$S_6Ѡv nP2$H{?[ AB3^P2$[NY7ЖQP2$V:fG/fP2$-Շ4RBiP2$lJDýuP2$go-)e piyP2$޶kdFzyPc $`33@337 )F ʚ;2Nʚ;g4:d:dv 0bppp0 <4!d!d` 0b5b<4dddd` 0b5b? %;  Outline  tState of the art in symbolic finite elements computation Semi-analytical/numerical approaches Enhancing finite element code performances Speeding up finite element code development Symbolic approaches and Object-Oriented programming for Finite Elements Hybrid Symbolic/Numerical Approach for Finite Elements: basic principles The FEM_Theory and FEM_Object approach The approach An example: A space  time formulation for the incompressible Navier-Stokes problem Constitutive law Modeling The object  Materiau Symbolic approach for constitutive law modeling Conclusions9|Hq`F 9|Hq ` F         r7>$Semi-analytical/numerical approachesB      Semi-analytical/numerical approaches Programming of the FEM in a symbolic computation environment: Maple, Mathematica& Enables us to keep symbolic values Example: Ioakimidis 1993 A parametric study in linear elasticity Principle: square domain, symmetric loads, 8 triangular elements Expansion in Taylor series Resolution of the linear system using Gauss-Seidel Method Solution in symbolic form with variable iteration numbers Rem: Completely symbolic 4 dofs !!!j%u%u  bj - o)Enhancing finite element code performance(   MEnhancing finite element code performance utilisation of a classical symbolic environment to enhance expressions by simplifying computations Example: Yang 1994 (see also Yagawa 1990) Numerical integration scheme Principle: Gauss quadrature: If f polynomial of degree 5: exact integration for n=3 48 multiplications and 17 additions Effectuate all algebraic operations: dependance on Gauss coordinates and material caracteristics 6 multiplications and 5 additions Extensions to fractions and to 2D computations Rem: Enhancement of the performances but completely problem dependent L*ZeZ*ZZ ZLZ$ZcZ#Z/ZZCZ*e*     .      $  c#/C~* w4  "SB+Speeding up finite element code developmentP       Automation of the development of FE models: automatic generation of code to compute elemental contributions Example: Noor 1981 (see also Wang 1981) Development of stiffness contributions for a thick shell element  Total Langrangian formulation  Displacement and mixed formulations (Hellinger-Reissner) Principle: Displacement formulation  Total potential energy: With: Finite element model Program in MACSYMA: entering numerical values: number of dofs, shape functions FORTRAN code for evaluation of the different finite element contributions Rem: generation for given elemental contributions, treatment of a new problem3 *N3 * Nwc 3 FQ]KGSymbolic approaches and Finite Elements Codes Advantages and Drawbacks .(1) S.A./N. Complete development of the FE model in a single and homogeneous environment Parametric studies: influence of parameters (geometric, materials& ) Applicable to small problems (less than 100 Dofs) Extension to NL problems difficult (2) Enhancement of FE codes performances Usefulness of algebraic computation systems to optimize the FE codes (in term of speed) Possibility to automatically generate FE code ( bug free !!) Only local action on the FE code (3) Speeding up FE code development Time spent to develop FE code drastically decreased Automatic production of FE code: Lack of flexibility, difficult to integrate new FE formulation Heavy systems in general (see FINGER  P.S. Wang 1981): multiple systems, and langages For a new problem, all the process is to be repeated ZZ)ZZ$Z Z V)"$U  ,  5 IHybrid Symbolic/Numerical Approach for Finite Elements: basic principlesj         High potential of symbolic approaches Taking advantages: wide range of problems, systematic approach for FE developments Avoid drawbacks: size of problems and extension to NL problems Idea: integrate the strong, weak and discrete forms of the FEM into a computerized environment AND use a classical FE code Power of symbolic concepts Efficiency of numerical codes Integration in natural way for the numerician (mechanics, mathematics& ) Problem: How to easy integrate these concepts in a computer? Integration in an OO environment Structuring of data simplified Hierarchy: codes more compact Natural way of programming&}=!X&}=! X  NHybrid Symbolic/Numerical Approach for Finite Elements: FEM_Theory/FEM_Object~           A Space-Time Formulation For Incompressible Navier-Stokes Formulation  Approximated Form+( *A numerical example: dam break problem (1)N        *A numerical example: dam break problem (2)N         Conclusions   fSur les dveloppements prsents& Seuls les aspects lis aux formulations E.F ont t abords Aspects constitutifs, aspects algorithmiques de base Efficacit des codes E.F. avec ces approches Problme du choix TECHNIQUE des outils utiliss (SMALLTALK/C++ ; Gibiane/FORTRAN-ESOPE). Les ingrdients pour aller plus loin& Concepts Intgrer au maximum les dveloppements dans un environnement informatis afin d viter une rupture dans les dveloppements d outils de simulation Technique Environnement homogne tant au niveau conceptuel que programmation Une seule approche informatique: paradigme objet Portabilit et distribution aise: LE WEB Efficacit du code acceptable & et JAVA!!!!!"Z<Z5ZZ&Z ZZ ZCZ1ZIZZ"<5& C1I&    Conclusions   Down: Only aspects linked to the formulations taken into account Constitutive aspects, algorithms,& Problems linked to the crucial choice of the tools (portability& ) Efficiency of the code generated Up: Homogeneous environment A single computerized approach: Object-Oriented paradigm Easy extendability to new formulations No preliminary hand studyv;#d;#d# 0'Interest of an Object-Oriented ParadigmJObject-Oriented programming Hierarchical organization of classes with inheritance Encapsulation of data and behavior inside the Object Communication between objects by sending messages (polymorphism) Application of the O.O.P. to the FEM Enhancement of the modularity of FE codes Better maintenability of FE codes Better extendability of FE codes Natural representation of algorithms and easy manipulation of elemental contibutions BASIC IDEA Today, one can integrate in existing finite element codes symbolic manipulations concepts linked to special problematic aspects of the FEMZZ%Z*ZCZYZ ZZ%*CY H T  A Space-Time Formulation For Incompressible Navier-Stokes Formulation  Strong FormB  A      7Finite Element code generated for Navier-Stokes STF (1)! 7Finite Element code generated for Navier-Stokes STF (2)B       /r n  ՜.+,0|    Format A4 (210x297mm)rormGD Times New Roman WingdingsMonotype SortsCentury SchoolbookModle par dfautMicrosoft Equation 3.0 EquationDocument Microsoft WordMicrosoft Word Document Outline%Semi-analytical/numerical approaches*Enhancing finite element code performance,Speeding up finite element code developmentHSymbolic approaches and Finite Elements Codes Advantages and Drawbacks JHybrid Symbolic/Numerical Approach for Finite Elements: basic principlesOHybrid Symbolic/Numerical Approach for Finite Elements: FEM_Theory/FEM_ObjectZA Space-Time Formulation For Incompressible Navier-Stokes Formulation Approximated Form+A numerical example: dam break problem (1)+A numerical example: dam break problem (2)<Implantation Interactive de Lois Constitutives en McaniqueObjet matriau*Approche symbolique pour loi constitutive Conclusions Conclusions(Interest of an Object-Oriented ParadigmTA Space-Time Formulation For Incompressible Navier-Stokes Formulation Strong Form8Finite Element code generated for Navier-Stokes STF (1)8Finite Element code generated for Navier-Stokes STF (2)Prsentatio