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 . Figure 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 0 -1 0 0 0 -1 0 A -1 -1 4 0 0 0 0 0 0 0 1 An Introduction to the Finite-Element Method ỐÓ9 and the vector b - The solution to the equation Ac b is 71 7c 7s 74 -75- which gives the following approximation to the solution of Laplace s equation and the boundary conditions on the respective triangles Tỵ. ộ x y 1 - 5 4 5y 4 -2 4 10 4 4 2 - 5 - 5y r2 ộ x ỳ 2 4 5 4 5y 4 4 - 10 4 4 -l 4 5 - 5y T3 ộ x ỳ 4 1 4 5y 4 4 2 5 5y 4 5 r4 ộ y 1 - 5 4 5y 4 -2 4 5 4 5y 4 2 - lOy T5 ộ x ỳ 2 - 5 4 5y 4 -4 4 10 4 4 3 - 5 - 5y rố 4 x y 6 10 4 6 4 10 4 lOy 4 4 1 lOy Tỵ. 4 x y 4 5 4 5y 4 5 4 4 1 5y Ts y 5y 4 4 1 5 4 4 5 5y T9 ộ x ý 10y 4 4 2 5 5y 4 4 1 4- 5 5y T10 ộ x ỳ 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 compares the value of u to the value of Ộ at Eị for each i 1 . 5. E Table y y u x y 670 CHAPTER 12 Numerical Solutions to .
