Mathematical Physics - Volume II - Numerical Methods

3.5 Finite element approximations

73

    K 2 11     0 K 0 0 0 0 0 0 0 0 0 0 K 2 21 0 K 2 22 K 2 23 0 0 0 K 2 31 0 K 2 32 K 2 33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K 6 11 0 K 6 12 K 6 13 0 0 0 0 0 0 0 0 0 0 K 6 21 0 K 6 22 K 6 23 0 0 0 K 6 31 0 K 6 32 K 6 33 0 0 0 0 0 0 0 0 2 12 K 2 13

    F     F

    f 2 1 0 f 2 2 f 2 3 0 0 0     0 0 f 6 1 0 f 6 2 f 6 3 0

    (3.114)

K 2 =

2 =

   

K 6 =

6 =

3. We establish the global equation system:

   

   

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

    =

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 1 F 2 F 3 F 4 ˜ F 5 ˜ F 6 F 7

    −

    Σ 1 0 0 Σ 4 Σ 5 Σ 6 0

   

(3.115)

=

where F 1 = f 3 2 · · · , and Σ i is defined based on (3.101). Terms denoted by ∼ will be modified when taking into account boundary conditions at Γ 56 . 4. Since non-homogeneous boundary conditions are defined only on the segment which connects nodes 5 and 6, γ and P have the following form: 1 1 + f 2 1 ; F 2 = f 1 2 + f

    0 0 0 0 γ 5 γ 6 0

    P =

   

    ,

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 P 55 P 56 0 0 0 0 0 P 65 P 66 0 0 0 0 0 0 0 0

γ =

(3.116)