Mathematical Physics - Volume II - Numerical Methods

Chapter 3. Comparison of finite element method and finite difference method

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hence the linear algebraic equation system (3.115) becomes:

   

   

    u 1 u 2 u 3 u 4 u 5 u 6 u 7

    =

   

    (3.117)

F 1 − Σ 1 F 2 F 3 F 4 − Σ 4

K 11 K 12 K 13 K 14

0 0 0

K 21 K 22 K 23 0 0 K 31 K 32 K 33 K 34 K 35 K 36 K 37 K 41 0 K 43 K 44 0 0 K 47 0 K 52 K 53 0 K 55 K 56 0 0 0 K 63 0 K 65 K 66 K 67 0 0 K 73 K 74 0 K 76 K 77 0 K 25

F 5 F 6 F 7

where K 55 = ˜ K 55 + P 55 , K 56 = ˜ K 56 + P 56 , F 5 = ˜ K 5 + γ 5 . 5. We can now introduce essential boundary conditions u 1 = u 4 = 0„ which reduces the linear algebraic equation system (3.117) to a system of equations with five unknowns:

  

  

   u 2 u 3 u 5 u 6 u 7

   =

   F 2 F 3 F 5 F 6 F 7

   (3.118)

K 22 K 23 K 25 0 0 K 32 K 33 K 34 K 36 K 37 K 52 K 53 K 55 K 56 0 0 K 63 K 65 K 66 K 67 0 K 73 0 K 76 K 77

which is solved for unknown displacements u 2 , u 3 , u 5 , u 6 and u 7 . Remaining two equation Σ 1 and Σ 4 : − Σ 1 = K 12 u 2 + K 13 u 3 + K 14 u 4 − F 1 , − Σ 4 = K 43 u 3 + K 47 u 7 − F 4 . (3.119)

Other solution properties can be determined based on the completely known uh, equation (3.86).

3.5.1 Determining of the finite element matrix

The first step involves the defining of a finite element mesh, followed by determining of necessary matrices. It should be taken into account that necessary matrices and vectors are very complex to determined when using a global coordinate system ( x , y ) , fig. 3.16, and that each finite element would then require individual determining of interpolation boundaries. Thus, matrices and vectors are determined in the local coordinate system ( ξ , η ) , and are then transformed into the global system. This allows for a unique procedure for all finite elements, making this task significantly easier.