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Χαράλαμπος Διδασκάλου
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1 208 5 [, 2, 3, 4, 5, 6, 7, 8] 2 () ϕ = λ θ () ϕ [W/m 2 ] θ [K] λ [W/(m K)] Schmatic rprsantation of Fourir s law (2) ρc θ = ϕ + f (2) ρ [kg/m 3 ] c [J/(kg K)] θ = t f [W/m3 ] () (2) (3) ρc θ = (λ θ) + f (3)

2 2 Schmatic rprsantation of th physical domain Numann N Diricht D (4) (5) = N D (4) N \ D = (5) Numann N q Diricht D θ (9) (7) q = λ θ on N (6) θ = θ 0 on D (7) 3 (3) (3) δθ (8) δθ ρc θ d = δθ (λ θ) + f} d (8) (8) (9) δθ ρc θ d = δθ (λ θ)d + δθ f d (9) (ϕa) = ( ϕ) A + ϕ A A = α ψ (0) (ϕ α ψ) = ( ϕ) α ψ + ϕ α ψ (0) (0) () ϕ α ψ = (ϕ α ψ) ( ϕ) α ψ () () (9) (2) δθ ρc θ d = (δθ λ θ) ( δθ) λ θ} d + δθ f d (2) (2) (3) δθ ρc θ d + ( δθ) λ θ d = (δθ λ θ) d + δθ f d (3) 2

3 (4) (ϕ ψ) d = (4) (3) (5) δθ ρc θ d + δθ λ θ d = n (δθ λ θ) d + n (ϕ ψ) d (4) δθ f d (5) (5) (6) n (δθ λ θ) d = δθ λ n d = δθ q d (6) (5) (6) (7) δθ ρc θ d + δθ λ θ d = (8) FEM = n δθ f d + δθ q d (7) () (8) (8) (9) (20) d d () (9) () d d () (20) () ϕ () ξ N ϕ (2) i ϕ () (ξ) = N(ξ) ϕ (7) (22) = N i (ξ) ϕ i (2) (N i δθ i ) t ρc N j θj d () + ( N i δθ i ) t λ N j θ j d () = () () (N i δθ i ) t f i d () + (N i δθ i ) t q i d () (22) () () (22) δθ i t (23) δθ t () N i t ρc N j θj d () + δθ t δθ t () N i t f i d () + δθ t 3 () ( N i ) t λ N j θ j d () = () N i t q i d () (23)

4 (23) δθ t (24) () N i t ρc N j θj d () + (24) (25) () N i t ρc N j d () θ + () N i t f i d () + () N i t f i d () + () ( N i ) t λ N j θ j d () = () N i t q i d () (24) () ( N i ) t λ N j d () θ = (26) (27) (28) () N i t q i d () (25) M = N t i ρc N j d () (26) () K = ( N i ) t λ N j d () (27) () f = N t i f i d () + N t i q i d () (28) () () (25) (26) (27) (28) (29) M θ + Kθ = f (29) 4 (29) Crank- Nicolson (30) t + t (29) M θ t+ t + Kθ t+ t = f t+ t (30) t + t θ t+ t (3) α 0 α θ t+ t = α θ t+ t + ( α) θ t (3) t + t f t+ t (32) f t+ t = α f t+ t + ( α) f t (32) t + t θ θ t+ t = t+ t t (33) θ t+ t = t+ t t = θ t+ t θ t t (33) 4

5 (3) (32) (33) (29) (34) M θ t+ t θ t t + K α θ t+ t + ( α) θ t } = α f t+ t + ( α) f t (34) (34) (35) M θ t+ t θ t t + Kθ t+α t = f t+α t (35) α = 0 Forward Eulr α = Crank-Nicolson α = Backward 2 Eulr Forward Eulr M Backward Eulr t Crank-Nicolson O( t 2 ) O( t) Crank-Nicolson α = (34) (36) 2 ( t M + 2 K) θ t+ t = 2 (f t+ t + f t ) + ( t M 2 K) θ t (36) 5 5. (35) FrontISTR (35) (37) t M(θ t+ t (i ) ) θ t+ t + K(θ t+α t (i ) ) θ t+α t = f(θ t+α t (i ) ) + t M(θ t) θ t (37) θ (i ) t+ t θ t+ t θ t+ t = G(θ(i ) t+ t ) (38) θ t+ t () = G(θ t+ t (0) ) θ t+ t (2) = G(θ t+ t () ) θ t+ t (3) = G(θ t+ t (2) ) (37). (38) 5

6 5.2 Nwton-Raphson (35) Nwton-Raphson φ(θ) (39) α = 2 φ(θ t+ t ) = t M(θ t+ t) + } 2 K(θ t+ t) θ t+ t + t M(θ t) + } 2 K(θ t) θ t 2 f(θ t+ t) + f(θ t )} = 0 (39) θ t+ t (40) θ t+ t (i+) = θ t+ t + (40) (40) (39) (4) φ(θ t+ t ) = t M(θ t+ t) + } 2 K(θ t+ t) θ t+ t + t M(θ t+ t) + } 2 K(θ t+ t) + t M(θ t) + } 2 K(θ t) θ t 2 f(θ t+ t) + f(θ t )} = 0 (4) M(θ t+ t ) K(θ t+ t ) f(θ t+ t ) (42) (43) (44) M(θ (i+) t+ t ) M(θ t+ t ) + M(θ t+ t ) (42) K(θ (i+) t+ t ) K(θ t+ t ) + K(θ t+ t ) (43) f(θ (i+) t+ t ) f(θ t+ t ) + f(θ t+ t ) (44) 6

7 (40) (42) (43) (44) (4) (45) } φ(θ t+ t ) = t M(θ t+ t ) + M(θ t+ t ) } + 2 K(θ t+ t ) + K(θ t+ t ) } + t M(θ t+ t ) + M(θ t+ t ) } + 2 K(θ t+ t ) + K(θ t+ t ) + t M(θ t) + } 2 K(θ t) θ t θ t+ t θ t+ t 2 f(θ t+ t ) + f(θ t+ t ) θ t+ t 2 f(θ t) = 0 (45) (46) φ(θ t+ t ) = t M(θ t+ t ) θ t+ t + M(θ t+ t ) θ t+ t + 2 K(θ t+ t ) θ t+ t + K(θ t+ t ) θ t+ t + t M(θ t+ t ) + 2 K(θ t+ t ) t M(θ t) θ t + 2 K(θ t) θ t 2 f(θ t+ t ) + f(θ t+ t ) θ t+ t 2 f(θ t) = 0 (46) (46) (47) (48) (49) M T (θ t+ t ) = M(θ t+ t ) θ t+ t K T (θ t+ t ) = K(θ t+ t ) θ t+ t F T (θ t+ t ) = f(θ t+ t ) (46) (47) (48) (49) (50) t M(θ t+ t ) + M T (θ t+ t ) + } 2 K(θ t+ t ) + K T (θ t+ t ) + F T (θ t+ t ) = t M(θ t+ t ) θ t+ t t M(θ t) θ t + 2 K(θ t+ t ) θ t+ t + 2 K(θ t) θ t 2 f(θ t+ t ) 2 f(θ t) (47) (48) (49) (50) = 0 (36) 7 (47) (48) (49) (50)

8 5.3 (5) (52) n M(θ) t ρ(θ k ) θ = N i c N j θ k d () () + N t i ρ c(θ k) N j θ k d () (5) () K(θ) n θ = ( N i ) t λ(θ k) () N j θ k d () (52) [] Klaus-JürgnBath, Mohammad R. Khoshgoftaar. Finit nt formulation and solution of nonlinar hat transfr. Nuclar Enginring and Dsign, Vol. 5, No. 3, pp , 979. [2] E.L. Wilson, K.J. Bath, and F.E. Ptrson. Finit nt analysis of linar and nonlinar hat transfr. Nuclar Enginring and Dsign, Vol. 29, No., pp. 0 24, 974. Spcial Issu: Paprs Prsntd at th Confrnc Contnts. [3] G.AGUIRRE RAMIREZ, J. T. ODEN. Finit nt tchniqu applid to hat conduction in solids with tmpratur dpndnt thrmal conductivity. Intrnational Journal for Numrical Mthods in Enginring, Vol. 7, No. 3, pp [4] NN Kochina. On a solution of th nonlinar diffusion quation. Journal of Applid Mathmatics and Mchanics, Vol. 28, No. 4, pp , 964. [5] A Srdar Slamt and B Murat Uzun. A novl and fficint finit nt softwar for hat transfr: Fhat. [6] Michal W Glass, Roy E Hogan Jr, and David K Gartling. Coyot: a finit nt computr program for nonlinar hat conduction problms. part i, thortical background. Tchnical rport, Sandia National Laboratoris, 200. [7] G Comini, S Dl Guidic, RW Lwis, and OC Zinkiwicz. Finit nt solution of non-linar hat conduction problms with spcial rfrnc to phas chang. Intrnational Journal for Numrical Mthods in Enginring, Vol. 8, No. 3, pp , 974. [8] L d F Frnch. Transint nonlinar hat transfr analysis using th finit nt mthod in th contxt of th rquirmnts of thrmal analysis in a min. PhD thsis, Univrsity of Cap Town,

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