tailieunhanh - Practical Ship Hydrodynamics Episode 12

Tham khảo tài liệu 'practical ship hydrodynamics episode 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Boundary element methods 211 ộxx -30x x - Xq - 0 r2 ộxy -30x y - Jq r2 ộxz -30x z - z r2 ộyy -30y y - Jq - 0 r2 0yz -30y z - z r2 ộxxz - 2 0 r2 C 50xx dz r2 ộxyz -50xy dz r2 ộxzz -50xz dz r2 - 30x r2 0yyz - 2 0 r2 C 50yy dz r2 0yzz 5ộyz dz r 3ộy r Regular first-order panel 1. Two-dimensional case For a panel of constant source strength we formulate the potential in a local coordinate system. The origin of the local system lies at the centre of the panel. The panel lies on the local x-axis the local z-axis is perpendicular to the panel pointing outward. The panel extends from x d to x d. The potential is then ln a x - x 2 C z2 dx With the substitution t x xq this becomes ln t2 C z2 dt x-d 2w 4n t ln t2 C z2 C 2z arctan - 2t 4M z Jx-d Additive constants can be neglected giving o . r1 . z X . . . 2dz . . Ộ lx ln C d ln r1r2 C z2 arctan 2-----2 2 C 4d with r1 x C d 2 C z2 and r2 x d 2 C z2. The derivatives of the potential still in local coordinates are 0x 1ln - 2 0z r1 r2 2dz Ơ 2b arctan x2 C z2 - d2 Ơ 2n 212 Practical Ship Hydrodynamics ộxx i .x. ộxxz ộxzz ơ í x C d x d 2h ri Ĩ2 ơ í 1 1 2n r1 r2 ơ 2 x- -d x d ã - -2 -1 -PT ĩ ơ x d 1 z2 x d 1 z 2 1 r2 r2 0x cannot be evaluated is singular at the corners of the panel. For the centre point of the panel itself 0z is _ . . Ơ 0z 0 0 D lim 0z 0 z D z 0 2 If the ATAN2 function in Fortran is used for the general expression of 0z this is automatically fulfilled. 2. Three-dimensional case In three dimensions the corresponding expressions for an arbitrary panel are rather complicated. Let us therefore consider first a simplified case namely a plane rectangular panel of constant source strength Fig. . We denote the distances of the field point to the four corner points by r1 D ựx2 C y2 C z2 r2 D ự x - 2 C y2 C z2 r3 D ựx2 C y - h 2 C z2 r4 D ự x - f 2 C y - h 2 C z2 Figure Simple rectangular flat panel of constant strength orgin at centre of panel The potential is ơ fh fz 1 D- I _ _ _ dệd n 0 0 y x - ệ 2 C y 2 C z2