7. Choice of approximating for the FE-method scalar problems

Transcription

1 7. Choice of approimating for the FE-method scalar problems

2 Finite Element Method Differential Equation Weak Formulation Approimating Functions Weighted Residuals FEM - Formulation

3 Weak form of heat flow

4 Approimating Functions Choose degree of approimation Choose polnomials Higher degree => higher accurac Number of terms = number of nodes Eamples of approimations: ( ) 3 ( ) (, )

5 Approimating functions, -dim Formulate approimation as function of geometr and nodal values ( ) C-matri method Lagrange interpolation polnomial ( ) ( ) ( ) L L Shape funct. N Shape funct. N Nodal value Nodal value

6 () () Approimating functions, -dim. Linear approimation ( ) Write on matri form ( ) Insert nodal values ( ) ( ) Solve for α α C a Insert eq. () in eq.() ( ) e he C-matri method Nα ; ; ; e a Cα e e e e e e N N N NC a N a L () L= - L

7 Approimating functions, -dim. Shape functions N e (linear functions) e e e ( ) NC a N a e e e e ( ) N N N N N e () N e L N e L= - L N e N e e N N L L e

8 Approimating functions N N i i 0 in node i in all other nodes

9 Approimating functions, -dim. N e i N e i L j i j N e j N e j L i i j

10 Approimating functions in -dim. weak form Approimate the temperature in each element b e e ( ) N a Weak form contains a derivative

11 Approimating functions Introduce matri B e in -dim. weak form and we can write For the one-dimensional element or if using the C-matri method

12 Element shape functions Approimation for entire bod built up element b element

13 Global shape functions

14 Global shape functions

15 Global shape functions emperature at nodal points Global shape function matri Approimation can be written as Gradient is given b or as where

16 Approimating functions, -dim. Formulate approimation as function of geometr and nodal values ( ) C-matri method Lagrange interpolation polnomial ( ) ( ) ( ) L L Shape funct. N Shape funct. N Nodal value Nodal value

17 Approimating Functions Lagrange s interpolation formula Cubic element approimation Lagrange s interpolation formula Set k = current shape function n = number of nodal points n- = degree of polnomial

18 Approimating functions, -dim.

19 Write the approimation as Reformulate on the form where Approimating functions, -dim. 3 ), ( k e k j e j i e i N N N ), ( ), ( N e are the nodal temperatures are the shape functions k j i,,

20 Write the approimation as Approimation at the nodal points Approimation ma be written Approimating Functions, -dim. α N k j i ), ( Cα k j i k k j j i i k j i e k j i k j i a C C Inverse: C-matri method e e e k j i a N a NC C ), (

21 Approimating Functions -dim. Inverse of C Determine area b Shape functions for -dim triangle element

22

23 Approimating Functions -dim. Gradient of the temperature that ma be written and we can write where

24 Approimating Functions -dim. For triangle element with three shape functions Inserting shape function gives B e is a constant matri for the triangle element

25 Global shape functions -dim. -dim. triangle elements

26 Summar

27 Convergence criteria Convergence criteria = When elements are infinitel small the approimate solution is infinitel close to the eact solution

28 Completeness (sv. fullständighet)

29 Compatibilit (sv. kompatibilitet) wo neighbouring elements must have the same temperature variation along their common boundar

30 Compatibilit he approimation must be continuous over the element boundaries

31 Compatibilit Eample: 4-node rectangle element Four nodes four terms Shape functions as where Shape functions determined from Lagrange s formula

32 Compatibilit Stud two connected elements Check lines =const and =const Check approimation along =c Collect terms => Linear approimation along =c and two nodes Check for =c => Linear approimation along =c and two nodes c c c 4 3 ), ( c c c 4 3 ) ( ) ( ), ( c c c 4 3 ) ( ) ( ), ( Element compatible! 4 3 ), (

33 Compatibilit Stud two connected elements Check approimation along =a+b Collect terms => quadratic approimation along =a+b and two nodes ) ( ) ( ), ( 4 3 b a b a b a ) ( ) ( ), ( a b a b b a Element NO compatible! =a+b 4 3 ), (

34 Rate of Convergence How fast the solution gets closer to the eact solution with decreasing element size. Linear elements:, = α + α + α 3 + Cr Quadratic elements, = α + α + α 3 + α 4 + α 5 + α 6 + Cr 3 where Cr n is the error of order n and r a distance he error e = C r n Linear elements: quadratic convergence, e = C r Quadratic elements: cubic convergence, e = C r 3

35 Choosing polnomial Choose a complete polnomial Avoid parasitic terms eamples 3 4 complete parasitic complete No parasitic! complete parasitic