10 Superelements and Global-Local Analysis IFEM Ch 10 Slide 1 - PDF document

Introduction to FEM 10 Superelements and Global-Local Analysis IFEM Ch 10 – Slide 1

Introduction to FEM Superelements Two extremes Macroelements "bottom up" Substructures "top down" IFEM Ch 10 – Slide 2

Introduction to FEM Substructuring was Invented in the Aerospace Industry (early 1960s) S4 S6 S2 S5 S1 S3 First level substructuring IFEM Ch 10 – Slide 3

Introduction to FEM Substructures (cont'd) level two substructure (wing section) level one substructure (wing) individual element IFEM Ch 10 – Slide 4

Introduction to FEM Multilevel FEM Substructuring was Invented in the Norwegian Offshore Industry in the mid/late 60s IFEM Ch 10 – Slide 5

Introduction to FEM Among Other Things, to Take Advantage of Repetition From DNV (Det Norske Veritas) web-posted brochure. Permission requested for inclusion in book proper. IFEM Ch 10 – Slide 6

Introduction to FEM Multistage Rockets Naturally Decompose into Substructures Short stack Apollo/Saturn lunar COMMAND MODULE rocket SERVICE MODULE ADAPTER LUNAR MODULE INSTRUMENT UNIT THIRD STAGE SIV-B IFEM Ch 10 – Slide 7

Introduction to FEM Static Condensation A universal way to eliminate internal DOFs b b i i b b b b b b b b Substructure Macroelement IFEM Ch 10 – Slide 8

Introduction to FEM Static Condensation by Matrix Algebra K bb K bi u b f b = Partition K ib K ii u i f i Solve for interior displacements from 2nd matrix equation u i = K −1 f i − K ib u b ( ) ii replace into first matrix equation ~ ~ Condensed K bb u b = f b stiffness equations ~ K bb = K bb − K bi K −1 where ii K ib ~ f b = f b − K bi K −1 ii f i IFEM Ch 10 – Slide 9

Introduction to FEM Static Condensation by Symmetric Gauss Elimination − 2 − 1 − 3 6 u 1 3       − 2 − 2 − 1 5 u 2 6 8 is called  =       − 1 − 2 − 4 7 u 3 4 the pivot      − 3 − 1 − 4 8 u 4 0 Task: eliminate u 4 ( − 3) × ( − 3) ( − 1) × ( − 3) ( − 4) × ( − 3) 0 × ( − 3) 6 − − 2 − − 1 − 3 −     u 1   8 8 8 8  = ( − 3) × ( − 1) ( − 1) × ( − 1) ( − 4) × ( − 1) 0 × ( − 1) − 2 − 5 − − 2 − 6 − u 2      8 8 8 8     ( − 3) × ( − 4) ( − 1) × ( − 4) ( − 4) × ( − 4) 0 × ( − 4) − 1 − − 2 − 7 − 4 − u 3 8 8 8 8 − 19 − 5 39 u 1 3       8 8 2  = Condensed − 19 − 5 39 6 u 2      8 8 2 equations − 5 − 5 5 u 3 4 2 2 IFEM Ch 10 – Slide 10

Introduction to FEM Static Condensation by Symmetric Gauss Elimination (cont'd) − 19 − 5 39 u 1 3       8 8 2  = − 19 − 5 39 u 2 6      8 8 2 − 5 − 5 5 u 3 4 2 2 Now eliminate u 3 ( − 5/2 ) � � u 1 � 3 − ( − 5/2 ) ( − 5/2 ) ( − 5/2 ) 4 × ( − 5/2 ) 8 − − 19 8 − 39 × × � � � 5 5 − 5 ( − 5/2 ) ( − 5/2 ) ( − 5/2 ) × ( − 5/2 ) 4 × ( − 5/2 ) − 19 8 − 8 − 6 − u 2 39 × 5 5 5 � 5 � � u 1 − 29 29 � � � Condensed = 8 8 − 29 29 equations u 2 8 8 8 IFEM Ch 10 – Slide 11

Introduction to FEM Static Condensation Module (posted on web site) CondenseLastFreedom[K_,f_]:=Module[{pivot,c,Kc,fc, n=Length[K]}, If [n<=1,Return[{K,f}]]; Kc=Table[0,{n-1},{n-1}]; fc=Table[0,{n-1}]; pivot=K[[n,n]]; If [pivot==0, Print["CondenseLastFreedom:", " Singular Matrix"]; Return[{K,f}]]; For [i=1,i<=n-1,i++, c=K[[i,n]]/pivot; fc[[i]]=f[[i]]-c*f[[n]]; For [j=1,j<=i,j++, Kc[[j,i]]=Kc[[i,j]]=K[[i,j]]-c*K[[n,j]] ]; ]; Return[Simplify[{Kc,fc}]] ]; ClearAll[K,f]; K={{6,-2,-1,-3},{ -2,5,-2,-1},{ -1,-2,7,-4},{-3,-1,-4,8}}; f={3,6,4,0}; Print["Before condensation:"," K=",K//MatrixForm," f=",f//MatrixForm]; {K,f}=CondenseLastFreedom[K,f];Print["Upon condensing DOF 4:", " K=",K//MatrixForm," f=",f//MatrixForm]; {K,f}=CondenseLastFreedom[K,f];Print["Upon condensing DOF 3:", " K=",K//MatrixForm," f=",f//MatrixForm]; IFEM Ch 10 – Slide 12

Introduction to FEM Static Condensation Module Results on Notes Example � � � � 6 � 2 � 1 � 3 3 � � � � � � � � � � � � � � � � � � � � � � 2 5 � 2 � 1 � � 6 � � � � � � � � � Before condensation: K � f � � � � � � � � � � � � � � 1 � 2 7 � 4 4 � � � � � � � � � � � � � 3 � 1 � 4 8 0 � � 39 � 19 � 5 � � � � ����� � ����� � ��� � � � � � 3 � 8 8 2 � � � � � � � � � � � � � � � � � 19 39 � 5 � � � Upon condensing DOF 4: K � � � f � � 6 � � � � ����� � ����� � ��� � � � 8 8 2 � � � � � � � 4 � 5 � 5 � � 5 ��� � ��� � 2 2 � � 29 � 29 � � � � ����� � ����� � � f � � 5 � � 8 8 8 � � � Upon condensing DOF 3: K � � � � � 29 29 � � ����� � ����� � 8 8 IFEM Ch 10 – Slide 13

Introduction to FEM Global-Local Analysis (an instance of Multiscale Analysis) Example structure: panel with small holes IFEM Ch 10 – Slide 14

Introduction to FEM Standard (one-stage) FEM Analysis finer meshes coarse mesh IFEM Ch 10 – Slide 15

Introduction to FEM Global-Local (two-stage) FEM Analysis Global analysis with a coarse mesh, ignoring holes, followed by local analysis of the vicinity of the holes with finer meshes (next slide) IFEM Ch 10 – Slide 16

Introduction to FEM Local Analysis BCs of displacement or (better) of force type using results extracted from the global analysis IFEM Ch 10 – Slide 17