热流科学与工程西幸文交通大堂教育部重点实验室Numerical Heat TransferChapter3NumericalMethodsforSolvingDiffusionEquationandtheirApplications(2)(Chapter4ofTextbook)QInstructorTao,Wen-QuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryof ThermalScience&EngineeringXi'anJiaotongUniversityXi'an,2022-Sept-281/56
1/56 Instructor Tao, Wen-Quan Key Laboratory of Thermo-Fluid Science & Engineering Int. Joint Research Laboratory of Thermal Science & Engineering Xi’an Jiaotong University Xi’an, 2022-Sept-28 Numerical Heat Transfer Chapter 3 Numerical Methods for Solving Diffusion Equation and their Applications (2) (Chapter 4 of Textbook)
热流科学与工程西幸文交通大堂E教育部重点实验室Contents3.11-D Heat Conduction Equation3.2FullyImplicit Scheme ofMulti-dimensionalHeat Conduction Eguation3.3TreatmentsofSourceTermand B.C3.4TDMA&ADIMethodsforSolvingABEs3.5FullyDevelopedHTinCircularTubes3.6FullyDevelopedHTinRectangleDucts2/56
2/56 3.1 1-D Heat Conduction Equation 3.2 Fully Implicit Scheme of Multi-dimensional Heat Conduction Equation 3.3 Treatments of Source Term and B.C. Contents 3.4 TDMA & ADI Methods for Solving ABEs 3.6 Fully Developed HT in Rectangle Ducts 3.5 Fully Developed HT in Circular Tubes
热流科学与工程西幸交通大堂E教育部重点实验室3.4TDMA & ADIMethodsforSolving ABEs3.4.1TDMAalgorithm（算法）for1-Dconductionproblem1.General form of algebraic equations of 1-Dconductionproblems2.Thomasalgorithm3.Treatment of 1st kind boundary condition3.4.2ADlmethodforsolvingmulti-dimensionalproblem1.Introduction tothematrixof 2-Dproblem2.ADliterationofPeaceman-Rachford3/56
3/56 3.4 TDMA & ADI Methods for Solving ABEs 3.4.1 TDMA algorithm (算法) for 1-D conduction problem 3.4.2 ADI method for solving multidimensional problem 1. Introduction to the matrix of 2-D problem 1.General form of algebraic equations of 1-D conduction problems 2. ADI iteration of Peaceman-Rachford 2.Thomas algorithm 3.Treatment of 1st kind boundary condition
热流科学与工程西幸文交通大堂E教育部重点实验室3.4TDMA&ADIMethodsforSolvingABEqs3.4.1TDMAalgorithmfor1-Dconductionproblem1.General form of algebraic equations.of 1-D conductionproblemsThe ABEqs fora,T +a,T +...+aT +...+amTm=b (i=l,Ml)steady and unsteady(f >0) problems takethe following form(I+1)(I-N)apT,=aT+awTw+bi=lThe matrix (矩阵)Threeofthecoefficients is atriunknownsdiagonal（三对角）one.4/56
4/56 3.4 TDMA & ADI Methods for Solving ABEqs 3.4.1 TDMA algorithm for 1-D conduction problem 1.General form of algebraic equations. of 1-D conduction problems The ABEqs for steady and unsteady (f >0) problems take the following form The matrix (矩阵) of the coefficients is a tridiagonal (三对角) one . P P E E W W a T a T a T b 1 1 2 2 1 1 .+ . = 1, 1) i i M M a T a T a T b i M a T （ Three unknowns i I (I-1) I (I+1)
热流科学与工程西幸文交通大堂E教育部重点实验室2. Thomas algorithm(算法The numbering method of W-P-E is humanized(人性化),but itcannotbeacceptedbyacomputer!Rewrite above equation:AT =BT+ +CT-, +D, i=1,2,....M1 (a)End conditions: i=1,C=C,=0; i=Ml,B,-Bm=0(1)Elimination（消元)-Reducingtheunknownsateachlinefrom3to2Assuming the eq.afterelimination asT,-, = P-,T, +Qi-↓(b)Coefficient has been treated tol5/56
5/56 2. Thomas algorithm(算法) Rewrite above equation： End conditions：i=1, Ci=C1 =0; i＝M1, Bi=BM1 =0 (1) Elimination (消元)－Reducing the unknowns at each line from 3 to 2 Assuming the eq. after elimination as 1 1 , 1,2,. 1 AT BT CT D i M i i i i i i i (a) T P T Q i i i i 1 1 1 Coefficient has been treated to 1. (b) The numbering method of W-P-E is humanized (人性化), but it can not be accepted by a computer!
热流科学与工程西步文通大学E教育部重点实验室The purpose of the elimination procedure is to findthe relationships between P,, Q, with A,, B,, C,, D,Multiplying Eq.(b) by C, and adding to Eq.(a):AT = B,T+I +CT- + D,(a)(b)CT =C,P-T +CO-AT -C,P-T = B,T+I +D, +C,Qi-IB,D, +C,Qi-)Ti++ +Yielding, -C,P-A-C,P,P-,T,+Comparing with T,-, =6/56
6/56 The purpose of the elimination procedure is to find the relationships between Pi , Qi with Ai , Bi , Ci , Di: Multiplying Eq.(b) by Ci , and adding to Eq.(a)： AT BT C T D i i i i i i i 1 1 (a) (b) Ci T P i 1 1 1 C C i i T Q i i i AT C P T i i i i i 1 BT D C Q i i i i i 1 1 1 1 1 1 ( ) i i i i i i i i i i i i B D C Q T T A C P A C P Comparing with T P T Q i i i i 1 1 1 Yielding
热流科学与工程西幸文通大堂E教育部重点实验室B,D, +C,Qi--C,P-1A, -C,Pi-1The above equations are recursive(递归的)一i.e.,In order to get P, Qi, P, and Q, must be known.In order to get Pj, Qr, use Eq.(a)AT =BT+ +CT- +D, i=1,2,....M1 (a)and the left end condition: i-l, C,=-0Applying Eq.(a) to i=1, and comparing it with Eq.(b)T,-1 = P-,T, + Q,-the expressions of Pj, Q, can be obtained:7/56
7/56 1 ; i i i i i B P A C P 1 1 ; i i i i i i i D C Q Q A C P The above equations are recursive (递归的)－i.e., In order to get Pi , Qi , P1 and Q1 must be known. 1 1 , 1,2,. 1 AT BT CT D i M i i i i i i i (a) and the left end condition：i=1, Ci =0 In order to get P1，Q1，use Eq.(a) Applying Eq.(a) to i=1, and comparing it with Eq.(b) the expressions of P1，Q1 can be obtained: T P T Q i i i i 1 1 1
热流科学与工程西青文交通大堂E教育部重点实验室From i=1,C, = O0, Eq.(a) becomes: AT = B,T, + D+DB,BiT.DAAAN(2)Backsubstitution(回代)-StartingfromM1via（顺序地）Eq.(b) to get T, sequentiallyB,TM=PMTMI+I+QM1, P=A -C,P-End condition:PMi = 0i=Ml,B=0to get:TM.-....T2,TTMI =QMIT-, = P-,T, +Qi-18/56
8/56 1 i C 1, 0, AT BT D 1 1 1 2 1 1 1 1 2 1 1 B D T T A A 1 1 1 ; B P A 1 1 1 D Q A (2) Back substitution(回代）－Starting from M1 via Eq.(b) to get Ti sequentially（顺序地） 1 1 1 1 1 , T P T Q M M M M End condition： i＝M1, Bi ＝0 T Q M M 1 1 1 ; i i i i i B P A C P 1 0 PM T P T Q i i i i 1 1 1 to get： 1 1 2 1 ,. , . T T T M From Eq.(a) becomes:
热流科学与工程西专交通大学E教育部重点实验室3.Implementation of Thomas algorithmfor1stkind B.C.For 1st kind B.C., the solution region is from i-2...toM1-1=M2, because T, and Tm are known.Applying Eq.(b) to i=1 with given T1,given : → P = 0; Q = Ti,givenT = PT +OBecause Tmis known, back substitution should bestarted from M,:TM2 = PM2TM1 +Q,When the ASTM is adopted to deal with B.C. of2nd and 3rd kind, the numerical B.C. for all cases isregarded as 1st kind, and the above treatment should beadopted.9/56
9/56 3. Implementation of Thomas algorithm for 1st kind B.C. For 1st kind B.C., the solution region is from i=2.to M1-1=M2, because T1 and TM1 are known. Applying Eq.(b) to i =1 with given T1,given： T PT Q 1 1 2 1 1P 0 ; Q T 1 1, given Because TM1is known，back substitution should be started from M2： T P T Q M M M 2 2 1 2 When the ASTM is adopted to deal with B.C. of 2 nd and 3rd kind, the numerical B.C. for all cases is regarded as 1st kind, and the above treatment should be adopted
热流科学与工程西青文交通大堂C教育部重点实验室3.4.2 ADl method for solving multi-dimensionalproblem1.Introductionto thematrixof 2-DproblemN1-Dstorage(一维存储）ofvariablesanditsrelationtomatrix coefficients10/56
10/56 3.4.2 ADI method for solving multi-dimensional problem 1. Introduction to the matrix of 2-D problem W P E N S S W P N E 1-D storage (一维存储）of variables and its relation to matrix coefficients