the FE-method

7. Choice of approximating for the FE-method

– scalar problems

Finite Element Method

Differential Equation

Weak Formulation Approximating

Functions

Weighted Residuals

FEM - Formulation

Weak form of heat flow

Approximating Functions

• Choose degree of approximation

– Choose polynomials

– Higher degree => higher accuracy – Number of terms = number of nodes

• Examples of approximations:

x x

T( ) ₁ ₂

3 4 2

3 2

) 1

(x x x x

T     y x

y x

T( , ) ₁ ₂ ₃

Approximating functions, 1-dim

• Formulate approximation as function of geometry and nodal values

x x

T( ) 





• C-matrix method

• Lagrange interpolation polynomial

2 1

1

2 1 ( )

) 1 (

)

( x x T

T L x

L x x

T     

Shape funct. N₁ Shape funct. N₂ Nodal

value

Nodal value

Approximating functions, 1-dim.

– The C-matrix method

• Linear approximation

• Write on matrix form

• Insert nodal values

• Solve for

• Insert eq. (2) in eq.(1) x

x

T( ) ₁ ₂

 



 



 





2 1 2

1 1

)

( 

 

 x x

x T

1 2 1

1 1)

(x T x

T   

2 2 1

2 2)

(x T x

T    1 ^;

; 1

2 1 2

1 2

1 



 



 



 



 



 







 x

x T

T a^e  Cα

ae

α  C^1 (1)

(2)

; )

(x ^{1 e e e}

T  NC^ a  N a ^

 

L x x L

x N x

N^{e e}

e 2 1

2

N 1

x T

T₂

T₁

x₁ x₂

T(x)

L=x₂-x₁

α

Approximating functions, 1-dim.

• Shape functions N

(linear functions)

e e

x e

T( )  NC^1a  N a

 

 L

x x L

x N x

N^{e e}

e 2 1

2

N 1

 

2 1 2

) 1

( N T N T

T N T

N x

T ^{e e}   ^e  ^e



 



 

x N^e₁

1

x₁ x₂

x N^e₂

1

x₁ x₂

L x N₁^e x  ²





L x N₂^e  x  ¹

x T

T₂

T₁

x₁ x₂

T(x)

L=x₂-x₁

Approximating functions

0 1





i i

N

N

in node i

in all other nodes

Approximating functions, 1-dim.

L x N_i^e x  ^j





L x N^e_j x  ⁱ



x N^e_i

1

x_i x_j

x N^e_j

1

x_i x_j

Approximating functions in 1-dim. weak form

e

x e

T( )  N a

• Approximate the temperature in each element by

• Weak form contains a derivative

Approximating functions in 1-dim. weak form

• Introduce matrix B^e

• and we can write

• For the one-dimensional element

• or if using the C-matrix method

Element shape functions

Approximation for entire body built up element by element

Global shape functions

Global shape functions

Global shape functions

• Temperature at nodal points

• Global shape function matrix

• Approximation can be written as

• Gradient is given by

• or as

where

Approximating functions, 1-dim.

• Formulate approximation as function of geometry and nodal values

x x

T( ) 





• C-matrix method

• Lagrange interpolation polynomial

2 1 1

2 1 ( )

) 1 (

)

( x x T

T L x

L x x

T     

Shape funct. N₁ Shape funct. N₂ Nodal

value

Nodal value

Approximating Functions

– Lagrange’s interpolation formula

• Cubic element approximation

• Lagrange’s interpolation formula

• Set

k = current shape function n = number of nodal points n-1 = degree of polynomial

Approximating functions, 2-dim.

• Write the approximation as

• Reformulate on the form

• where

Approximating functions, 2-dim.

y x

y x

T ( , )  

 

 

k e k j

e j i

e

i T N T N T

N y

x

T( , )   

) , ( x y N

T

k j i ,,



• Write the approximation as

• Approximation at the nodal points

• Approximation may be written

Approximating Functions, 2-dim.

 



















k j i

y x y

x T





 1

) , (

 Cα



















































k j i

k k

j j

i i

k j i

y x

y x

y x

T T T





 1

1 1

e

k j i

k j i

T T T

a C C^1  ^1























































 Inverse:

C-matrix method

 

k j i

T T T y

x y

x

T C  NC a  N a

































 1 ^{1 1}

) , (

• Inverse of C

• Determine area by

• Shape functions for 2-dim triangle element

Approximating Functions – 2-dim.

Approximating Functions – 2-dim.

• Gradient of the temperature

that may be written

• and we can write where

Approximating Functions – 2-dim.

• For triangle element with three shape functions

• Inserting shape function gives

• B^e is a constant matrix for the triangle element

Global shape functions – 2-dim.

2-dim. triangle elements

Summary

Convergence criteria

• Convergence criteria = When elements are infinitely small the approximate solution is infinitely close to the exact solution

Completeness (sv. fullständighet)

Compatibility (sv. kompatibilitet)

• Two neighbouring elements must have the same

temperature variation along their common boundary

Compatibility

The approximation must be continuous over the element boundaries

Compatibility

• Example: 4-node rectangle element

• Four nodes four terms

• Shape functions as

• where

• Shape functions determined from Lagrange’s formula

Compatibility

• Study two connected elements

• Check lines x=const and y=const

• Check approximation along x=c

• Collect terms

=> Linear approximation along x=c and two nodes

• Check for y=c

=> Linear approximation along y=c and two nodes

cy y

c y

c

T( , ) ₁ ₂ ₃ ₄

y y

c c

y c

T( , )  (₁ ₂ )(₃ ₄ )  ₁  ₂

x x

c c

c x

T( , )  (₁ ₃ ) (₂ ₄ )  ₁  ₂

Element compatible!

xy y

x y

x

T( , ) ₁ ₂ ₃ ₄

Compatibility

• Study two connected elements

• Check approximation along y=ax+b

• Collect terms

=> quadratic approximation along y=ax+b and two nodes )

( )

( )

,

(x ax b _{1 2}x ₃ ax b ₄x ax b

T       

2 3 2

1

2 4

4 2

2 3

1 ) ( )

( ) ,

(

x x

ax x

b a

b b

ax x T







































Element NOT compatible!

y=ax+b

xy y

x y

x

T( , ) ₁ ₂ ₃ ₄

Rate of Convergence

• How fast the solution gets closer to the exact solution with decreasing element size.

• Linear elements:

𝑇 𝑥, 𝑦 = 𝛼₁ + 𝛼₂𝑥 + 𝛼₃𝑦 + C𝑟²

• Quadratic elements

𝑇 𝑥, 𝑦 = 𝛼₁ + 𝛼₂𝑥 + 𝛼₃𝑦 + 𝛼₄𝑥² + 𝛼₅𝑥𝑦 + 𝛼₆𝑦² + C𝑟³

• where C𝑟^𝑛 is the error of order n and r a distance

• The error e = C 𝑟 ^𝑛

Linear elements: quadratic convergence, e = C 𝑟 ² Quadratic elements: cubic convergence, e = C 𝑟 ³

Choosing polynomial

• Choose a complete polynomial

• Avoid parasitic terms

• examples

complete parasitic xy y

x

T ₁ ₂ ₃ ₄

complete No parasitic!

2 6 5

2 4 3

2

1 x y x xy y

T      

complete parasitic

y x xy

y xy

x y

x

T ₁ ₂ ₃ ₄ ² ₅ ₆ ² ₇ ² ₈ ²