Residual and convergence rate - University of Florida

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... Element Analysis. R.T. Haftka ... Finite Element Analysis. R.T. Haftka ... O(hp+1- r)in representing the r th derivative of the field quantity. O(h2(p+1 m))i ti. t i.

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9.5: Residuals • Because of numerical errors residual is not zero

R  R K D

• Error E measure (what ( h t does d it represent?) t?) e 

D T  R  D T R 

• Iterative improvement Ri  R K Di K Di  Ri

Di 1  Di  Di

• Why is it attractive?

R.T. Haftka

EML5526 Finite Element Analysis

University of Florida

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Error analysis

q 0 • Example, E l rod d element l t u,xx  AE

R.T. Haftka

EML5526 Finite Element Analysis

University of Florida

3

Error bounding • Displacement Di l error   x  xi e  x   u  x   ui 1  hi  

  x  xi   ui 1    hi

  

• Let z be the point in the element where e’=0 x

x

z

z

e '  x   e '  z   e '  x    e ' ' s  ds   u ' ' s  ds

• Then x

 u ' ' s  ds  z

R.T. Haftka

x

 z

u ' ' s  ds 

x i 1



xi

u ' ' s  ds  hi  max u "  x    xi  x  xi 1 

EML5526 Finite Element Analysis

University of Florida

4

Bounds on displacement and strain errors • Error on strains e '  x   hi  max u"  x    xi  x  xi1 

• With some more algebra g 1 2 e  x   h i  max 8  x i  x  x i 1

R.T. Haftka

u "  x   

EML5526 Finite Element Analysis

University of Florida

5

General features • Definitions h = approximate “characteristic length” of element: length of a linear element; length of longest line segment that fits within a plane or solid element (one option) p = degree of highest complete polynomial in the element field quantity 2m = order of the highest derivative of the field quantity in the governing differential equation • Then errors are: – O(hp+1)in representation of the field quantity – O(hp+1-r)in representing the r th derivative of the field quantity 2(p+1 m))in – O(h2(p+1-m) )i representing ti strain t i energy R.T. Haftka

EML5526 Finite Element Analysis

University of Florida

6

9.7 Multimesh extrapolation • Let L t O(hq) be b th the order d off error iin phi hi 1h2q  2 h1q 1  2 ( h1 / h2 ) q   or   q q q h2  h1

R.T. Haftka

1  ( h1 / h2 )

EML5526 Finite Element Analysis

University of Florida

7

Regular mesh refinement

R.T. Haftka

EML5526 Finite Element Analysis

University of Florida

8

Graphical representation • Reduced integration with hour-glass control outperforms exact or plain reduced integration.

R.T. Haftka

EML5526 Finite Element Analysis

University of Florida

9

Irregular mesh refinement

•Straight St i ht liline fitt fitted d to t three th data d t points i t R.T. Haftka

EML5526 Finite Element Analysis

University of Florida