Numerical MethodsThe Nodal NetworksFinite Difference ApproximationFinite Difference Approximation (cont.)Slide 5A System of Algebraic EquationsMatrix FormNumerical SolutionsIterationExampleExample (cont.)Slide 12Numerical MethodsDue to the increasing complexities encountered in the development of modern technology, analytical solutions usually are not available. For these problems, numerical solutions obtained using high-speed computer are very useful, especially when the geometry of the object of interest is irregular, or the boundary conditions are nonlinear. In numerical analysis, two different approaches are commonly used: the finite difference and the finite element methods. In heat transfer problems, the finite difference method is used more often and will be discussed here. The finite difference method involves: Establish nodal networks Derive finite difference approximations for the governing equation at both interior and exterior nodal points Develop a system of simultaneous algebraic nodal equations Solve the system of equations using numerical schemesThe Nodal NetworksThe basic idea is to subdivide the area of interest into sub-volumes with the distance between adjacent nodes by x and y as shown. If the distance between points is small enough, the differential equation can be approximated locally by a set of finite difference equations. Each node now represents a small region where the nodal temperature is a measure of the average temperature of the region.Example:m,nm,n+1m,n-1m+1, nm-1,nyxm-½,nintermediate pointsm+½,nx=mx, y=nyFinite Difference Approximation2P21Heat Diffusion Equation: ,kwhere = is the thermal diffusivityCNo generation and steady state: q=0 and 0, 0tFirst, approximated the first order differentiation at intermediateq TTk tVT     &&1, ,( 1/ 2, ) ( 1/ 2, ), 1,( 1/ 2, ) ( 1/ 2, ) points (m+1/2,n) & (m-1/2,n)TxTxm n m nm n m nm n m nm n m nT TTx xT TTx x          Finite Difference Approximation (cont.)21/ 2, 1/ 2,2,21, 1, ,2 2,2Next, approximate the second order differentiation at m,n/ /2( )Similarly, the approximation can be applied to the other dimension ym n m nm nm n m n m nm nT x T xTx xT T TTx xT         , 1 , 1 ,2 2,2( )m n m n m nm nT T Ty y  Finite Difference Approximation (cont.)2 21, 1, , , 1 , 1 ,2 2 2 2,22 2( ) ( )To model the steady state, no generation heat equation: 0This approximation can be simplified by specify x= yand the nodal m n m n m n m n m n m nm nT T T T T TT Tx y x yT                 1, 1, , 1 , 1 ,equation can be obtained as4 0This equation approximates the nodal temperature distribution based onthe heat equation. This approximation is improved when the distancem n m n m n m n m nT T T T T       between the adjacent nodal points is decreased:Since lim( 0) ,lim( 0)T T T Tx yx x y y           A System of Algebraic Equations• The nodal equations derived previously are valid for all interior points satisfying the steady state, no generation heat equation. For each node, there is one such equation.For example: for nodal point m=3, n=4, the equation isT2,4 + T4,4 + T3,3 + T3,5 - 4T3,4 =0T3,4=(1/4)(T2,4 + T4,4 + T3,3 + T3,5)• Nodal relation table for exterior nodes (boundary conditions) can be found in standard heat transfer textbooks. (ex. F.P. Incropera & D.P. DeWitt, “Introduction to Heat Transfer”.)• Derive one equation for each nodal point (including both interior and exterior points) in the system of interest. The result is a system of N algebraic equations for a total of N nodal points.Matrix Form11 1 12 2 1 121 1 22 2 2 21 1 2 2The system of equations:N NN NN N NN N Na T a T a T Ca T a T a T Ca T a T a T C         LLM M M M MLA total of N algebraic equations for the N nodal points and the system can be expressed as a matrix formulation: [A][T]=[C]11 12 1 1 121 22 2 2 21 2 = , ,NNN N NN N Na a a T Ca a a T Cwhere A T Ca a a T C                               LLM M M M M MLNumerical SolutionsMatrix form: [A][T]=[C]. From linear algebra: [A]-1[A][T]=[A]-1[C], [T]=[A]-1[C]where [A]-1 is the inverse of matrix [A]. [T] is the solution vector.• Matrix inversion requires cumbersome numerical computations and is not efficient if the order of the matrix is high (>10)• Gauss elimination method and other matrix solvers are usually available in many numerical solution package. For example, “Numerical Recipes” by Cambridge University Press or their web source at www.nr.com.• For high order matrix, iterative methods are usually more efficient. The famous Jacobi & Gauss-Seidel iteration methods will be introduced in the following.Iteration11 131 1 32 2 33 3 1 11( ) ( ) ( 1)1General algebraic equation for nodal point:,(Example : , 3)Rewrite the equation of the form:i Nij j ii i ij j ij j iN Niij ijk k kii j jj j iii ii iia T a T a T Ca T a T a T a T C ia aCT T Ta a a            L1N• (k) - specify the level of the iteration, (k-1) means the present level and (k) represents the new level.• An initial guess (k=0) is needed to start the iteration.• By substituting iterated values at (k-1) into the equation, the new values at iteration (k) can be estimated• The iteration will be stopped when maxTi(k)-Ti(k-1), where  specifies a predetermined value of acceptable errorReplace (k) by (k-1)for the Jacobi iterationExampleSolve the following system of equations using (a) the Jacobi methos, (b) the Gauss Seidel iteration method.4 2 112 0 32 4 16X Y ZX Y ZX Y Z       ,* ,(a) Jacobi method: use initial guess X0=Y0=Z0=1, stop when maxXk-Xk-1,Yk-Yk-1,Zk-Zk-1  0.1First iteration: X1= (11/4) - (1/2)Y0 - (1/4)Z0 = 2Y1= (3/2) + (1/2)X0 = 2Z1= 4 - (1/2) X0 - (1/4)Y0 = 13/4Reorganize into new form:X =114-12Y -14ZY =32+12X + 0 * ZZ = 4 -12X -14Y4 2 1 111 2 0 32 1 4 16XYZ                     