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Burden - Numerical Analysis 5e (PWS, 1993) Episode 3 Part 8

Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) episode 3 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Ĩ 668 CHAPTER 12 Numerical Solutions to Partial-Differential Equations u x y 4 for x y e Le and x y G L7 T x y X for x Ji e L2 and x y gL4 dn du z T- x y y ơn for x y e L5 dw v X x y dn V2 for x ỹ G Lỵ and x Ji E L3 where dw dn denotes the directional derivative in the dữection of the normal to the boundary of the region D at the point x y . We first subdivide D into triangles with the labeling suggested in Step 0 of the algorithm. For this example Lg kJ Li and 5 2 L UÍ2 uf3 uf4 u L5. The labeling of triangles is shown in Figure 12.16. Figure 12.16 The boundary condition u x y 4 on L6 and L7 implies that yf 4 when Í 6 7 . 11. To determine the values of 7 for I 1 2 . 5 apply the remaining steps of the algorithm and generate the matrix 2.5 0 -1 0 0 0 1.5 -1 -0.5 0 A -1 -1 4 0 0 0 -0.5 0 2.5 -0.5 0 0 0 -0.5 1 12.4 An Introduction to the Finite-Element Method ỐÓ9 and the vector 6.0666 b - 0.0633 8.0000 6.0566 2.0316 The solution to the equation Ac b is 71 4.0383 7c 4.0782 7s 4.0291 74 4.0496 -75- 4.0565 which gives the following approximation to the solution of Laplace s equation and the boundary conditions on the respective triangles Tỵ. ộ x y 4.0383 1 - 5 4 5y 4 4.0291 -2 4 10 4 4 2 - 5 - 5y r2 ộ x ỳ 4.0782 2 4 5 4 5y 4 4.0291 4 - 10 4 4 -l 4 5 - 5y T3 ộ x ỳ 4 1 4 5y 4 4 2 5 5y 4 4.0383 5 r4 ộ y 4.0383 1 - 5 4 5y 4 4.0782 -2 4 5 4 5y 4 4.0291 2 - lOy T5 ộ x ỳ 4.0782 2 - 5 4 5y 4 4.0496 -4 4 10 4 4 3 - 5 - 5y rố 4 x y 4.0496 6 10 4 4.0565 6 4 10 4 lOy 4 4 1 lOy Tỵ. 4 x y 4 5 4 5y 4 4.0383 5 4 4 1 5y Ts y 4.0383 5y 4 4 1 5 4 4 5 5y T9 ộ x ý 4.0291 10y 4 4 2 5 5y 4 4 1 4- 5 5y T10 ộ x ỳ 4.0496 10y 4 4 3 5 - 5y 4 4 -2 4 5 - 5y . The actual solution to the boundary-value problem is u x y y 4 4. Table 12.7 compares the value of u to the value of Ộ at Eị for each i 1 . 5. E Table 12.7 y y u x y 0.2 0.2 4.0383 4.04 0.0017 0.4 0.2 4.0782 4.08 0.0018 0.3 0.1 4.0291 4.03 0.0009 0.5 0.1 4.0496 4.05 0.0004 0.6 0.1 4.0565 4.06 0.0035 670 CHAPTER 12 Numerical Solutions to .