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Heat Conduction Basic Research Part 15

Tham khảo tài liệu 'heat conduction basic research part 15', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Meshless Heat Conduction Analysis by Triple-Reciprocity Boundary Element Method 339 Numerical solutions are obtained using the interpolation functions for time and space. If a constant time interpolation and time step tk - tk-1 are used the time integral can be treated analytically. The time integrals for Tf p q t r are given as follows J rip qJr dr - 1 1 71 Jtf 4k7T j n OT1 p q t r 1 Or J . -----------dr NTTÌÍ7exp -af 72 Jtf On 2ktĩì On J where r2 af Wf 73 Assuming that functions T Q r and õT Q r On remain constant over time in each time step Eq. 65 can be written in matrix form. Replacing T Q r and õT Q r On with vectors Tf and Qf respectively and discretizing Eq. 65 we obtain Brebbia 1984 F F z HfFTf 2 GfFQf Bo f 1 f 1 74 where Bo represents the effect of the pseudo-initial temperature. Adopting a constant time step throughout the analysis the coefficients of the matrix at several time steps need to be computed and stored only once. If there is heat generation the following time integrals are used Ochiai 2001 . C.F__ . . - r . . 1__ _ . J T2 p q l T dr E1 af -Ị- E1 af ln af c 1 -exp -Of Jtf 16NĨT 1 tlf 1 1 1 75 .F qq . r d 0L 1 -exp af E1 af pF Jtf On dr 8ktt On Of 76 r 4 f3 p q t r dr 1 ._ . . 1 E1 Of 4E1 Of 4ln Of 4C 1 - exp -Of -y 2E Of 256kk a Of 2 ln Of 2C - 2a f 3 - 3exp -Of - 5af 77 tF OTi p q . r dr - - F 1 -exp 2Of -Of E1 af .f On 64kk On af2 2 1 . . . . . . 2E1 af 2ln af 2C 1 - exp af 78 af 340 Heat Conduction - Basic Research Additionally the temperature gradient is given by differentiating Equation 65 and expressed as . . _ T_ . _ . _. _ . . 9T p t -Kf f T Q t 9 - t -9T Q t 9T1 tMĩdn õXị j0 r õXịõn 9n õXị 2 f f t f 9Tf 1 p Q t T 9Wf Q t f 1 f0 fr 9Xi 9n 92T 1 p Q t T _ __ ----MM Wf QWfc dXịdn J Mot X.Í W3P dm t m 1 9T3 p Vm tT T 9Xi 2 1f9Tf 1 p Q t 0 9T f Q 0 - Jr iXr------- n - -9 á t 0 TfWWTdr -M -P Vm 0 9T3 p Vm t 0 d 79 dXịdn J m 1 9Xi The derivative of the polyharmonic function Tf P q t T and the normal .