Linear Quadratic and Cubic Interpolation Models-Finite Element Method-Assignment Solution, Exercises of Mathematical Methods for Numerical Analysis and Optimization

Download Linear Quadratic and Cubic Interpolation Models-Finite Element Method-Assignment Solution and more Exercises Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!

ASSIGNMENT #0 5

NUMERICAL

PROBLEM # 4. 27

NUMERICAL PROBLEM # 4.

STATEMENT

Consider a ring element with a triangular cross-section as shown in the figure. If the field variable does not change with respect to Ɵ, propose linear, quadratic and cubic interpolation models for Co^ continuity. Develop the necessary element equations for the linear case for solving the Laplace’s equation

2 2 2 2

1 0 r r r z

          

TO FIND

a) Linear, quadratic and cubic interpolation models for Co^ continuity b) Necessary element equations for the linear case for solving the Laplace’s equation

SOLUTION

a) Linear, quadratic and cubic interpolation models for Co^ continuity

i) Linear interpolation model

   1   2 x  3 y

ii) Quadratic interpolation model

2 2

   1   2 x   3 y   4 x   5 xy  6 y

ii) Strain in terms of nodal displacements and shape functions

rr

zz rz



    

 

rr e zz rz

u r u r (^) B Q w z u w z r



    (^)     ^    ^    ^   (^)    (^)           (^)    (^)     (^)    ^  

 

0 0 0

1 0 0 0 tan 2 0 0 0

i j k

i j k

i j k i i j j k k

b b b

N N N B (^) r r r cons t A c c c c b c b c b

              

iii) Stress vector

  (^)  D (^)   (^)  D  (^) B Q  e  cons tan t

rr

zz rz



iv) Element stiffness matrix

e

e T V

 (^) K    B D B dv

e

e T V

 (^) K  (^)  (^) B  D B (^)  dv

V  2  rA

 K e^   BT  D  B  2  rA

v) Consistent load vector

a) ( )  

e i^ e^ T o V

P  (^)  B D  dv

e i e^ T T V

P B D E  dv



i e^ T T

P E^ ^ B  rA

b)  

e

e^ T b V

P  (^)  N  dv

0 0 0 0 0 0 i i e j r b A j z k k

L L L P L L L

 

  ^ 

  ^ 



r  r Li i  r Lj j  r Lk k            

i j k^ r i j k z e i^ j^ k r b i j k z i j k r i j k z

r r r r r r A r^ r^ r P r r r r r r r r r

 ^  

 ^  

 ^  