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

Macroelements "bottom up" Substructures "top down"

Two extremes

IFEM Ch 10 – Slide 2

Introduction to FEM

Substructuring was Invented in the Aerospace Industry (early 1960s)

S1 S2 S3 S4 S5 S6

First level substructuring

IFEM Ch 10 – Slide 3

Introduction to FEM

Substructures (cont'd)

level one substructure (wing) level two substructure (wing section) 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 rocket

COMMAND MODULE SERVICE MODULE ADAPTER LUNAR MODULE INSTRUMENT UNIT THIRD STAGE SIV-B

IFEM Ch 10 – Slide 7

Introduction to FEM

Static Condensation

b b b b b b b b b b i i

Substructure Macroelement A universal way to eliminate internal DOFs

IFEM Ch 10 – Slide 8

Introduction to FEM

Static Condensation by Matrix Algebra

Kbb Kbi Kib Kii ub ui = fb fi ui = K−1

ii

fi − Kibub Kbbub = fb Kbb = Kbb − KbiK−1

ii Kib

fb = fb − KbiK−1

ii fi

Solve for interior displacements from 2nd matrix equation replace into first matrix equation where

Condensed stiffness equations

Partition

( )

~ ~ ~ ~

IFEM Ch 10 – Slide 9

Introduction to FEM

Static Condensation by Symmetric Gauss Elimination

Task: eliminate u

4

 

39 8

− 19

8

− 5

2

− 19

8 39 8

− 5

2

− 5

2

− 5

2

5     u1 u2 u3   =   3 6 4      6 − 2 − 1 − 3 − 2 5 − 2 − 1 − 1 − 2 7 − 4 − 3 − 1 − 4 8       u1 u2 u3 u4    =    3 6 4   

   6 −

(− 3)×(− 3) 8

− 2 −

(− 1)×(− 3) 8

− 1 −

(− 4)×(− 3) 8

− 2 −

(− 3)×(− 1) 8

5 −

(− 1)×(− 1)

− 2 −

(− 4)×(− 1) 8

− 1 −

(− 3)×(− 4) 8 8

− 2 −

(− 1)×(− 4) 8

7 −

(− 4)×(− 4) 8

     u1 u2 u3   =    3 −

0×(− 3) 8

6 −

0×(− 1) 8

4 −

0×(− 4) 8

  

8 is called the pivot Condensed equations

IFEM Ch 10 – Slide 10

Introduction to FEM

Static Condensation by Symmetric Gauss Elimination (cont'd)

• 39

8 − (− 5/2 ) (− 5/2 ) 5

− 19

8 − (− 5/2 ) (− 5/2 ) 5

− 19

8 − (− 5/2 ) (− 5/2 ) 5 39 8 − (− 5/2 )× × × × (− 5/2 ) 5

u1 u2

• −

3 −

4×(− 5/2 ) 5

6 −

4×(− 5/2 ) 5

• 29

8

− 29

8

− 29

8 29 8

u1 u2

• =

5 8

• Now eliminate u3

 

39 8

− 19

8

− 5

2

− 19

8 39 8

− 5

2

− 5

2

− 5

2

5     u1 u2 u3   =   3 6 4   Condensed equations

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

Before condensation: K

• 6

2 1 3 2 5 2 1 1 2 7 4 3 1 4 8

• f
• 3

6 4

• Upon condensing DOF 4: K
• 39
• 8

19

• 8

5

• 2

19

• 8

39

• 8

5

• 2

5

• 2

5

• 2

5

• f
• 3

6 4

• Upon condensing DOF 3: K
• 29
• 8

29

• 8

29

• 8

29

• 8
• f 5

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

coarse mesh finer meshes

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

BCs of displacement or (better) of force type using results extracted from the global analysis

Local Analysis

IFEM Ch 10 – Slide 17