DELFT UNIVERSITY OF TECHNOLOGY

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1 DELFT UNIVERSITY OF TECHNOLOGY REPORT 8-3 Numerical Metods for Industrial Flow Problems Ibraim, C. Vuik, F. J. Vermolen, and D. Hegen ISSN Reorts of te Deartment of Alied Matematical Analysis Delft 8

2 Coyrigt 8 by Deartment of Alied Matematical Analysis, Delft, Te Neterlands. No art of te Journal may be reroduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mecanical, otocoying, recording, or oterwise, witout te rior written ermission from Deartment of Alied Matematical Analysis, Delft University of Tecnology, Te Neterlands.

3 NUMERICAL METHODS FOR INDUSTRIAL FLOW PROBLEMS IBRAHIM, C. VUIK, F. J. VERMOLEN, AND D. HEGEN Abstract. In tis reort, Partial Differential Equations PDEs are solved numerically. Tese include Poisson Equation, Convection-Diffusion Equation, eiter steady state or time-varying, and Burgers Equation. Metods used are Standard Galerkin Algoritm SGA and Streamline Uwind Petrov-Galerkin. For te nonlinear Partial Differential Equations, Picard Iteration is combined wit Finite Element Metods. For time-varying PDEs, Elicit, Imlicit and IMEX scemes are used. A system of nonlinear equations is solved by alying te above mentioned metods.. Introduction Matematical modeling of many ysical systems is often described by so called conservation equations. In general, suc equations are reresendted by nonlinear couled artial differential equations. Analytical solutions are ard or imossible to find for many PDEs. Eerimental data are not available all te time. Numerical Metods can be used even in suc cases and sometimes it is te only coice. Error estimation, solution convergence and stability are imortant issues in numerical metods.. Finite Element Metod Te finite element metod is a numerical tecnique, used to find aroimate solutions of artial differential equations. It is more general, as comared to te finite difference and te finite volume metods in te sense tat, it can andle comle geometries and it is not restricted to differential equations in conservative form. Te main features of te finite element are as follows. Te domain of te roblem is reresented by a collection of simle subdomains, called elements. Te collection of tese elements is called te finite element mes. Over eac element, a roerty u described by a artial differential equation is aroimated and reresented by a sum of olynomials wit coefficients u j, wic are found from a set of algebraic equations... Standard Galerkin Aroimation. We will use an eamle to elain te standard Galerkin metod. Consider te Poisson equation in one dimension: D d u = f,, u =, u =. Multily equation by a test function v and integrate it over te comutational domain, [,. d u D v = fv. 3 To get te weak formulation, aly integration by arts. [ du D v du dv + D = fv. 4 Te autor is indebted to HEC, Pakistan and NUFFIC, Te Neterlands for teir financial and oter suort.

4 We coose v suc tat integrals in equation 4 eist. Furtermore, we take v = and v =. Terefore, te boundary term in te above equation vanises. Now we discretize, u and v in te following way. Te comutational domain [, is divided into n finite intervals by considering n + nodes, =,, 3,..., n =. Eac suc interval is called an element. All elements are taken wit equal lengt = n. Te it element e i is te interval from i to i. Te unknown variable u is aroimated by u = {u j }, were u j u j, and j =,,..., n. Now u is written in terms of basis functions, φ j, as follows, u u j φ j, 5 were u = u and u n = u are known. We coose te function v as, j= v = φ i, i =,,..., n. 6 Now φ i is a iecewise linear function defined as: i i i, for [ i, i, φ i = i+ i i+, for [ i, i+,, for [ i, i+. 7 φ = {, for [,,, for [,.. φ n = { n n n, for [ n, n,, for [ n, n φ φ i φ n Figure. Te iecewise linear basis function: φ i φ i j = {, for j i,, for j = i. Wit tis discretization, equation takes te form or D j= u j n D j= u j dφ j dφ j = = fφ i, i =,,..., n, 8 fφ i + b i, i =,,..., n. 9

5 Were b i is te it element of vector b wic incororates boundary conditions and it is given as, D dφ u dφ, for i =, b i = D n dφ n u n dφ n n, for i = n,, < i < n, D u, for i =, D b i = u n, for i = n,, < i < n. All te integrals on te left-and side of equation 9 can be comuted analytically. For fφ i, we will use a numerical tecnique, if it is ard to determine oterwise. In order to imlement equation 9 to a comuter rogram, we use te following strategy. Equation 9 can be written in a matri form: Su = f + b, were S is an n by n matri, called te stiffness matri, u is te vector consisting of u j, j =,,..., n and f is a vector of lengt n. Te elements of S and f are given by, S i,j = D f i = dφ j, fφ i. In order to find te matri S, we calculate te element matri S e, [ i S e = D i i i, i i S e = D [ i i. For te current roblem, we ave D =, = =., n =, f =, u =, u n =. Initially S is an n + by n + matri filled wit zeros for all elements. Wit an inde k, running from to n, te stiffness matri S is udated as: S k,k = S k,k + S e,, S k,k = S k,k + S e,, S k,k = S k,k + S e,, S k,k = S k,k + S e,. Before using S in furter calculations, we omit its first and te last row as well as te first and te last column because we need to find only n variables, u i. For te integral on te rigt and side, we can use a Newton-Cotes formula te traezoidal rule, f f + f. 3 Of course, te integral in our case is simle enoug to calculate analytically. Now te element vector for f is given as: [ f e = e i fφ i. 4 e i fφ i 3

6 Again an inde k, running from to n, els to udate te vector f, initially loaded wit zeros only. f k = f k + f e, f k = f k + f e. Te first and te last elements of f are not considered. Hence we are left wit n elements. Now te aroimate solution vector u j can be determined, once we ave S, f and b, u = S f + S b. 5 Once we ave te element matri, in tis case S e, assembly of te corresonding matri S is te same for all roceeding eamles. Ecet tat, if a boundary is not subject to an essential boundary condition, we will no longer remove te corresonding row and column of te stiffness matri. A natural boundary condition alies in tis case. Te vector b and f are treated in a similar way. Te equation 5 as been imlemented in MATLAB. Te result is sown in Figure, along wit te relative difference wit te eact solution, u = 3 +. We observe tat for suc a simle roblem, tere is almost no error i.e., u = u j at node oints = j. In tis eamle, boundary conditions were secified elicitly for a Numerical Solution.8.6 u b Relative Error in Numerical Solution Figure. a Te solution of Poisson equation by SGA, D =, f =, u =, u =, b Relative Error u u j u j, u = 3 +. bot ends. We can ave oter tye of boundary conditions. Among many, few tyes are mentioned below. Diriclet Boundary Conditions: If boundary values are given elicitly, tey are called Diriclet Boundary conditions. e.g., u =, u = a etc. For our urose tey are also called Essential Boundary Conditions. Neumann Boundary Condition: It is te derivative of te unknown variable given at certain boundary oint, e.g., du = at =. Robin Boundary Condition: It is a combination of above two boundary conditions. e.g., du + σu = a, σ >, =. For second order artial differential equations, te last two conditions are also called natural boundary conditions. If we do not secify boundary condition for one end, say at =, ten du = for = alies by default [. Of course it is true only aroimately for a numerical solution. In tis case we no longer omit te last column and row from S, f and b. Furtermore b n = b n =. Te result for tis new condition is sown in Figure 3, wit u =. 4

7 .95 Solution u Aroimate Solution Eact solution > Figure 3. Te solution of te diffusion equation by red SGA, D =, f =, u =, =, blue Te eact solution, u = +. du 3. Convection Diffusion Problem In tis section, we considered te steady state convection diffusion equation. Initially, we alied Standard Galerkin metod to comute te aroimate solution. Wen convection dominates over diffusion, we face undersireable wiggles in te numerical solution. We alied anoter metod to overcome tis difficulty. ɛ d u + cdu =, < <, 6 u =, u =, were c is seed of te wave and ɛ is te diffusion coefficient. Bot c and ɛ are assumed to be constant. Function u could be te temerature or some oter ysical quantity. In order to solve tis equation, we aly te Standard Galerkin aroac. Multily equation6 by a test function v and integrate over te roblem domain. d u ɛ v + c du v =. 7 Integration by arts for te first integral on te left-and side gives te following result. [ du ɛ v du dv + ɛ + c du v =. Again, coosing v = v = makes te first term equal to zero. Discretization used for u is given by, u u j φ j, j= were u = u and u n = u, are known. Te definition of te basis functions φ j and v is te same as given in te revious section. Equation 7 can be written as, d ɛ u j φ j j= + c d 5 u j φ j φ i =, i =,,..., n. j= 8

8 Now te element stiffness matri is given by, [ i S e = ɛ i i +c i i [ i i i S e = ɛ i i i φ i i φ i [ i i i i + c [ φ i φ i, 9, were is te lengt of an element. We ave divided our domain uniformly, terefore is a constant. Te first element of te vector b is given by, b = ɛ u dφ dφ c u dφ φ, b = ɛ u + c u. Similarly, b n = ɛ u n c u n. All oter elements of b are zero. Equation 8 in te matri form is written as, Su = b, u = S b Tis equation as been imlemented in MATLAB. Two gras are sown in Figure 4. In te blue gra, ɛ =, c =, wile in te dotted gra, ɛ =., c = 4. Wen convection dominates over diffusion term, we observe wiggles in te outut, were te Standard Galerkin Aroimation is used. To get rid of tese undesired wiggles, we used a metod called Streamline Uwind Petrov-Galerkin SUPG..4 Solution u Equal Convection Diffusion Convection Dominant > Figure 4. Te numerical solution of te convection-diffusion equation, blue ɛ =, c =, red ɛ =., c = 4. 6

9 Streamline Uwind Petrov-Galerkin. In tis metod, a modified test function is used, w = v +, were v is te classical test function. Te additional term introduces artificial diffusion and ence counters te convection dominance. Tis idea is taken from te finite difference metod [. = ξ. For a first order uwind, we ave ξ =, wic we will use in tis study oter values of ξ are also used for different reasons. Hence, w = φ i +. Using w in equation 7 instead of v we ave, d ɛ u j φ j j= ɛ d du dw + c φ i + dφ i + c du w =, d u j φ j j= φ i + =. Te additional term of is effective only in te convection art since elementwisely, te derivative of is zero. Te element matri in tis case is given by, [ i S e = ɛ i i i [ i i i i i + c i φ i i i i i φ i i i [ + c i i i i i i S e = ɛ b takes te form, b = ɛ [ u dφ 3 dφ c + c [ i u dφ φ c b = ɛ u + c u + c u. + c [ u dφ i φ i. 4 dφ, Note tat te discretization of u, t is not canged. Now S and b are determined as usual and te equation is used for te numerical solution. Te solution obtained by te Streamline Uwind Petrov Galerkin SUPG metod is sown in Figure 5, along wit te eact solution, u = e e m e m, m = c m ɛ. Again, we used ɛ =., c = 4, but wit te test function w. We observe tat te numerical solution in tis case is smoot but it deviates from te analytical solution, were u varies sarly. Tis cliing effect is due to te artificial diffusion term and it can be reduced by taking a finer mes. Te result sown in Figure 6 indicates tis fact. We eect tat u u as n. Again, a finer mes means tat etra memory and more comutational time is required. For te current roblem, a finer mes also eliminates te need of SUPG metod. 7 φ i,

10 Solution u SUPG Eact Solution > Figure 5. Te solution of te convection-diffusion equation, ɛ =., c = 4, u =, u =, by SUPG metod wit =. and analytically, u = e m e m e m, m = c ɛ,. Solution u SUPG Eact Solution > Figure 6. Te solution of te convection-diffusion equation, ɛ =., c = 4, u =, u =, by SUPG metod wit =. and analytically, u = e m e m e m, m = c ɛ,. 3.. Solution of Transient Convection Diffusion Equation. Consider te following time-deendent convection-diffusion equation, u, t t u, t + c = D u, t. 5 u, t =, u, t = t, u, = e e +. Here D and c, reresent te diffusion coefficient and te flow velocity and tese quantities are assumed to be constant. We can use te Newton-Cotes rule in case c and D are secified by some given functions of. Initial conditions u, and boundary values u, t and u, t are also given. After multilying equation 5 by a test function v of our coice and integrating over te roblem domain, we 8

11 get te weak formulation. u, t u, t u, t v v + c v = D. 6 t We divide our domain into n elements as elained in te revious section, ten te discretization of u, t is done as follows u, t u j tφ j, were u t and u n t are given. j= j= For brevity, we will use te notation u j t = u j and φ j = φ j. n u j φ j φ i + c t n u j φ j φ i = D u j φ j φ i, n d dt were, j= u j j= n φ j φ i = c u j j= n dφ j φ i D j= u j j= i =,,.., n. dφ j + b i, u D dφ dφ + c dφ φ, for i =, b i = n u n n D dφ n dφ n + c dφ n φ n, for i = n,, < i < n, D u + u c for i =, D b i = u n u n c, for i = n,, < i < n. Te matri form of te equation8 is given by, M du dt were te i, jt element of te mass matri M is, M i,j = 7 8 = Su + b, 9 φ i φ j. Its element matri M e is given by, [ i M e = D i φ i φ i i i φ i φ i i i, 3 i φ i φ i i φ i φ i M e = [ 6. 3 Te rocedure for te assembly of te matri M is eactly te same as for te matri S. Alying te te Forward Euler sceme for te time discretization, we ave, M u+ u = Su + b,

12 were is te time inde. We ave taken a time ste =., ence =,, 3... corresonds to,, 3..., in te time scale. Equation 33 can be written as, u + = I M Su + M b. 34 Note tat u is already comuted in te revious iteration. Tis time-discretization sceme is called elicit. In te elicit sceme we are faced wit a stability condition. Let λ be an eigenvalue of I M S, ten sould be cosen suc tat, λ, λ. Oterwise we will ave an unstable result. For te current eamle we ave, =., =., n =, D =, u, t =, u, t = t, u, = e e e. Te result is sown in Figure 7. u,t Figure 7. Te numerical solution of te time deendent convection-diffusion equation by SGA, D =, c =, u, t =, u, t = t, u, = e e + Imlicit Sceme: If we use Backward Euler sceme for equation 3, we ave M u+ u = Su + + b +. Let G = M + S, ten, u + = GMu + Gb +. Tis equation is imlemented in MATLAB. Te result is not sown because it is visually te same as for te elicit case. An advantage of te imlicit metod is tat it is unconditionally stable. But, in general, it is relatively more demanding in terms of use of memory and rocessing time. 3.. Solution of te Non-Linear Burgers Equation by Finite Elements. Consider te following non-linear equation, also called Burgers Equation, u, t t u, t + u, t = D u, t, 35 u, t =, u, t = t, u, = Te weak formulation of equation 35 is given by, u, t v + t u, t u, t v = D e e e +. u, t v, t. 36

13 In tis equation te non-linear term u u is te difficult art. It can be treated by a number of ways. One otion is tat we use te same discretization for every u, given by, u, t u j tφ j, 37 j= were u t and u n t are known. Wit tis sceme, te given non-linear equation takes te form, u j φ j φ i + u j φ j u k φ k φ i t j= = D j= k= n u j φ j φ i, i =,,.., n, j= 38 n d dt j= u j n φ j φ i = D u j j= dφ j L i + b i, i =,,..., n, were L i results from te non-linear term and b i contains te boundary values. Comutation of L i, i.e., te discretized form of te non-linear term in te it row is given by, n L i = u k φ k u j φ j φ i, i =, 3.., n. 4 k= j= We will determine L and L n searately, because tey are sligtly different from te general form L i. Now considering only te nonzero terms in equation 4, we ave, L i = i i u i φ i + u i φ i i+ i u i φ i + u i+ φ i+ u i + u i u i + u i+ φ i + + φ i, L i = i u i + u i u i φ i + u i φ i φ i + i u i + u i+ L i = u i + u i u i 6 + u i Now L and L n are given by, i+ i 3 u i φ i + u i+ φ i+ φ i, + u i + u i+ u i L i = 6 [ u i u i u i + u i u i+ + u i+. L = 6 [ u u + u u + u, L n = 6 [ u n u n u n + u n u n. 3 + u i+ 6, 39

14 In L and L n, we did not consider 6 u and 6 u n. Tese known values are taken by b and b n, resectively, b = u 6 b n = u n Te matri form of equation 39 is given by, M du dt 6 + u D, + u n D. = Su L + b. After alying te Euler Backward sceme, we get, M + Su + = L + + Mu + b +, were is a time inde and L + = Lu +. In a Picard iteration, we already ave u +, eiter as an initial guess or a value close to te final solution, in case we ave convergence. Let G = M + S, ten, u + = G L + + Mu + b +. Let b = GMu + b +, wic is a constant for eac time iteration. u + = GL + + b. 4 Tis equation is solved by Picard Iteration. Picard iteration is used to solve nonlinear equations written in te form, u new = fu. 4 Were u starts wit an initial guess. We calculate u new by using equation 4. In te net iteration, te value of u new is assigned to u. Tis rocedure goes on until a stoing condition is satisfied. Te iteration stos successfully if te infinity norm of u new u u is less tan a secified value, i.e., new u ma new u < δ, u new were δ is an accetable error level. In tis case we move on to te net time ste. If tis condition is not met witin a secified number of iterations or u new, te Picard iteration stos, unsuccessfully. Equation 4 can be formulated in many ways. If it is divergent in one form, it migt converge in anoter form. For te current time deendent roblem, we used u as te initial guess in Picard iteration for u +. Tis rocedure is more elaborated in Algoritm. Te aroimate solution u j, is sown in Figure 8. In Algoritm, it can be seen tat te vector L is comuted from u +, but u + is taken from te revious Picard iteration. Hence te entire non-linear term is treated at te revious Picard iteration. Semi-Imlicit and Fully Imlicit Metods We reconsider te non-linear roblem equation 35, first discretize it in time and ten in sace. In tis way we take certain unknown variables at revious time iteration te semi-imlicit sceme, ence te roblem is linearized. u + u + u u+ = D u +,,, =,, 3,..., 43

15 u,t Figure 8. Te numerical solution of Burgers equation by SGA, D =, u, t =, u, t = t, u, = e e + Solution Algoritm for te Non-linear Burgers Equation n=number of elements, = n, j = j, j =,,..., n, u t =, u n t = t, u, = e e, Initialize L, b, M, and S wit zeros, Comute Me, Se, M, S, and G, Time Loo Starts wit =, u + = u, %serves as guess Comute b, b n, b, Picard Loo starts ere, Comute L, y = GL + b If stoing condition=true, break Picard Loo u + = y Picard Loo Ends If condition=divergence, break Time Loo Time Loo Ends Dislay results were u = u,. Te weak form and te corresonding sace-discretization are given by, u + u v + j= u + j u j φ j φ i + +D u, t u+ j= v = D n u k φ k k= u + j 3 φ j u + j= u + j φ j dv, φ i =, i =,,..., n

16 In te following comutations, we used a Newton-Cotes integration rule te Traezoidal rule given by equation 3. Te element stiffness matri is determined in te following way, S e, = i i u i φ i + u i φ i dφ i i φ i + D i, S e, = u i + D. Te comlete element matri is given by, S e = D [ + [ u i u i u i u i. 46 Since, S is a function of u, terefore, S e and S are comuted at eac time iteration. We will denote S by S. Te mass element matri M e is different in tis case, because we ave not comuted it analytically. M e = [ 47 Te mass matri M is assembled as usual. Te first element of te vector b is given by, b = u φ + u φ u + dφ φ D D b = u+ u + u+. Similarly, b n = u+ n u n + u+ n D. Now te equation 45 written in a matri form is given by, u + dφ dφ, M u+ u + S u + = b, 48 u + = M + S Mu + b. 49 Te numerical solution is sown in Figure 9. Now we consider te fully imlicit u,t Figure 9. Te numerical solution of Burgers equation by te semi-imlicit sceme, D =, u, t =, u, t = t, u, = e e + 4

17 case. In tis case, again Euler Backward sceme is used for discretization of te time derivative but all oter variables are taken at te current time iteration +. We used a Picard metod to solve te non-linear roblem, so it is linearized wit resect to Picard iteration. Te time-discretized form of equation 35 is given by, u + new u + u + unew + = D u + new,,, =,, 3,... 5 Te metod elained in Algoritm 3. is equivalent to te following linearization. S e = D u + u + new u + u + = D u + new,,, =,, 3,... 5 Te element stiffness matri corresonding to equation 5 is given by, [ [ u + i u + i, 5 + u + i u + i Similarly te matri form corresonding to te discretization of equation 5 is given by, M u+ new u + S + u + new = b +, 53 u + new = M + S + Mu + b Equation 54 is solved by Picard iteration. Te result is close to te semi-imlicit case. For te comarison, teir difference is sown in Figure. Te difference decreases wit te increasing time. Relative Difference Between Semi and Fully Imlicit Solutions Figure. Te difference in two solutions of Burgers equation by Semi-Imlicit and Fully-Imlicit scemes, D =, u, t =, u, t = t, u, = e e 4. System of Non-linear Equations In tis section, we ave comuted a numerical solution for a system of nonlinear, couled equations. Te system of equations is given by, ρ t + ρv T = D + q, t, 55 ρ t + ρv =, 56 5

18 v, t = λ P, 57 P, t.v = RT, t, 58, t = ct, t. 59 Tis is a one-dimensional, simlified case of te flow of a certain fluid troug a orous medium. Te system is taken from Master Tesis of Abdelaq Abouafç. In tis system, te unknown variables are te density ρ, t, te velocity v, t, te ressure P, te temerature T, and te entaly, t. On te oter and te volume V, te diffusion coefficient D, te universal gas constant R, λ, and c are given constants. Te source function q, t is also known. In tis section, te symbol and v are already taken, so we will use te symbol for te lengt of an element and η for te classical basis function. We ave used bot, te semi-imlicit and te fully imlicit sceme to solve tis roblem. After eliminating P and T, we rewrite te above system as, ρ t + ρv = D + q, t, 6 ρ t + ρv =, 6 v, t = λ, 6 were D = D /c and λ = λr cv. From equation 6 and 6 we ave, ρ t + ρ t + ρv + ρv = D + q, t, 63 ρ t = ρv. 64 Using equation 64 into equation 63, we ave, ρ t ρv + ρv + ρv = D + q, t, 65 ρ + ρv t = D + q, t. 66 Using v = λ in equations 66 and 6 we ave te following system of equations, ready to be solved numerically. ρ t λρ = D + q, t, 67 ρ t λ ρ λρ =. 68 Te initial conditions and te boundary values are given by,, = 3,, t = + sin5t,, t = + cost ρ, =.5, ρ, t =.5 Tese equations are linearized in te same manner as in case of Burgers equation i.e., not more tan one variable is taken at current time inde, in eac term semiimlicit case. Equations 67 and 68 take te forms wit q,t=, ρ + λρ 6 + = D +, 69

19 ρ + ρ λ ρ + λρ+ =. 7 Now te weak form of equation 69 and 7 is given by, ρ + η λ ρ + ρ + ρ w λ λ ρ + η = D ρ + + w We ave used te following discretization for unknown variables,, t i tφ i, ρ, t ρ i tφ i. i= i= Now te discretization of equation 7 is given by, ρ k φ k + j j φ jφ i λ ρ k φ k k= j= = D j= k= + φ j j dη, 7 w =. 7 l= l φ l j= + φ j j φ i i =,..., n. We will use tis equation to find te entaly. Terefore we add an subscrit to te matrices corresonding to tis equation. Te first element of te mass element matri M e, is given by, M e, = i 73 ρ i φ i + ρ i φ i φi φ i = i ρ i, 74 were we ave used a Newton-Cotes integration rule te traezoidal rule. Te comlete element matri M e is given by, M e = [ ρ i ρ. 75 i Similarly, te first element in te stiffness element matri S e, is determined as follows, i S e, = λ ρ i φ i + ρ i φ i ρ i i + dφi ρ i φ i D φ i, 76 S e, = λ Te comlete element matri S e is given by, S e = λ i + [ ρ i ρ i i ρ i ρ i i + i ρ i + D D [. 78 From tese element matrices, we assemble global matrices, M and S. Let b be a vect wic includes boundary values corresonding to variables in equation 69. 7

20 Ten b = S, and b n = S n,n n. Te element matrices corresonding to equation 7 are given as follows Streamline Uwind Petrov-Galerkin alied, M ρe = [, S ρe = λ i + [ i Te final equations in te matri form are given by, + = M + S M + b, ρ + = M ρ + S ρ Mρ ρ + b ρ.. 8 Te numerical solution obtained from tese equations is sown in Figure for two time resolutions =. and =.. Te remaining variables, v, P, and T are comuted as ost rocessing. In case of fully imlicit sceme, equation 67 and t =. ρ t =. ρ Figure. Te numerical solution of and ρ, D =, q =, =.,, = 3,, t = + sin5t,, t = + costρ, =.5, ρ, t =.5 68 are linearized in te following way q=, ρ + + new λ ρ new = D + new, 8 ρ + new ρ λ + ρ + new λ ρ + new + =. 8 8

21 Were te subscrit indicates te value at revious Picard iteration. Te element matrices are given by, M e = ρ + i. 83 S e = λ + i S ρe = λ + + i + i + ρ + i ρ + i ρ + i + i Te final equations in te matri from are given by, + M = + new + S + ρ + M = new ρ + Sρ + M + D [ ρ + i ρ + i [ M ρ ρ + b + ρ b + In Figure, te relative difference and ρ ρ ρ is sown, were and come resectively from semi-imlicit and fully imlicit solutions. A similar argument alies to ρ and ρ. Te difference is more noticeable were te gradient is large in actual variables.., / ρ ρ /ρ Figure. and ρ ρ ρ, D =, q =, =.,, = 3,, t = + sin5t,, t = + costρ, =.5, ρ, t =.5 5. Finite Elements in Two Dimensions We ave divided te two-dimensional unit domain into triangular elements. Te mes and te element toology is sown in Figure 3. Total number of elements in tis case is n = n n y, were n and n y are te node count in resective 9

22 directions. We ave considered -node boundary elements. Total number of boundary elements are n + n y = n + n y 4. For two-dimensional roblems trougout tis reort, we maintained tis convention for node and boundary numbering order. Γ Γ y Γ Γ Figure 3. A 55 mes, red numbers indicate nodes and blue number are te element labels. Γ i are te non-overlaing boundary segments Again, te linear basis functions φ as been used. Were = [, y T φ is defined as, and φ = a + b + cy, 86 φ i j = δ ij. 87 For a tyical element e k, wit nodes,, and 3, te basis function is determined by using equations 86 and 87, given by, a a a 3 b b b 3 y y =, 88 c c c 3 y were, k = [ k, y k T, k =,, 3. All te coefficients are determined by solving equation 88 and tey are given by, b = y y 3, b = y 3 y, b = y y, c = 3, c = 3, c =, a = b c y, a = b c y, a 3 = b 3 3 c 3 y Were = y 3 y y y 3, wic also is twice te area of te triangle. Hence we require non-zero area for all elements. For te triangular elements, te following Newton-Cotes quadrature rule as been used: gdω = 6 g + g + g 3. 9 Ω

23 5.. Poisson Equation in Two Dimensions. Te steady-state diffusion equation in two dimensions is given by,. D u = f, on Ω 9 u =, at Γ, u = +, at Γ 3, u ˆn =, on Γ Γ 4, were, =, y, and ˆn is te outward ointing unit normal vector. Te comutational domain is reresented by te oen region Ω, wile Γ = 4 i= Γ i reresents te domain boundary. Te weak form of equation 9 is given by te following equation.. D u ηdω = fηdω. 9 Using Gauss Teorem, we ave, D u. ηdω D Ω Ω Γ Ω u ˆn ηdγ = fηdω. 93 Ω We take η = for Γ Γ 3. Tis makes te boundary integral Γ ˆn ηdγ equal to zero because u ˆn = for Γ Γ 4. After eliminating te boundary integral, te discretized form of equation 93 is given by, D u j φj. φ i dω = fφ i dω, Ω j= Ω i {,,..., n} for wic i / Γ Γ Te element matri S e ertaining to te kt element e k is comuted as follows, S ei,j = D φj. φ i dω = 6 b ib j + c i c j, i, j =,, Ω Let k, k, and k3 be tree nodes of e k. Ten te stiffness matri S is udated in te following way, S ki,k j = S ki,k j + S ei,j, i, j =,, 3, k =,,..., n. A similar rocedure is followed for te rigt-and side of equation 94 i.e., te element vector f e is determined for all elements and te global vector f is udated accordingly. Equation 94 written in a matri form is given by, S n n u n = f n. 96 Net we artition vectors and te matri in te above equation suc tat all te known values in u n could be taken to te rigt and side of te equation. Let n = m + r suc tat u m are unknown values and u r are given values from essential boundary conditions i.e., we rearrange equation 96 for rows and columns for tis urose. Equation 96 is rewritten as, From tis equation we ave, S m+r m+r u m+r = f m+r, 97 S r m S r r ur [ Sm m S m r [ um [ f = m S m m u m + S m r u r = f m, f r u. 98

24 S m m u m = f m S m r u r, u = S f + b, were b = S m r u r y Figure 4. D =, f = Te numerical solution of te diffusion equation, 5.. Convection-Diffusion Equation in Two-Dimension. Given te roblem,. D u + v. u =, 99 u =, at Γ, u =, at Γ 3, u ˆn =, at Γ Γ 4. Let Se D and Se C be element matrices due to diffusion and convection terms resectively. We ave comuted Se D in te revious section and te oter art is given by, Se C i,j = v. φj φ i dω = Ω 6 v b j + v y c j, i, j =,, 3. Te numerical solution of equation 99 is sown in Figure 5. Now boundary u y Figure 5. Numerical Solution of Steady-State Convection- Diffusion equation, D =, n = n y = 5

25 .65 u,y, y = Figure 6. Gra of u, y, y =.75 eected to be constant, D =, n = n y = 5 conditions are suc tat all variations are in te y-direction and we eect a constant solution in te -direction. In Figure 6 we ave sown te solution u, y, y =.75. We observe a small variation even in te -direction. In order to investigate te cause of tis, we divide our domain into two elements, sown in Figure 7 and take te following boundary conditions, u =, at Γ, u ˆn =, at Γ 3, u ˆn =, at Γ Γ 4. Only two element matrices S e and S e are to be determined in tis case. Taking e e Figure 7. Comutational Domain wit two elements v = everywere, we ave te element matri due to te convection art, S C e i,j = 6 v yc j, i, j =,, 3, first element. It is clear from equation 89 tat c j can be eressed in terms of and. Hence we can write, S C e i,j = β j, c were is defined in Figure 7 and β j = v j y 6 is indeendent of te element dimension and te area because c j. Te comlete element matri for 3

26 te first element e is given by, S C e = β β β 3 β β β 3 β β β 3 = v y 6. 3 Wit current toology β is always zero regardless of and values. We udate te global matri S C and it is given by, β β 3 S C = β β 3. 4 β β 3 Similarly Se C and te udated S C are given by, Se C = β β 3 β β 3 = v y 6 β β 3 S C = Now we come to te diffusion art; β β 3 β 3 β β 3 β β 3 β β 3 β 3, 5. 6 S D e = b ib j + c i c j, i, j =,, 3. 7 From equations 89 and 7, it is obvious tat Se D is indeendent of and. Te diffusion art of te stiffness matri comuted from Se D and Se D is given by, S D =, 8 wic is symmetric about bot diagonals. Terefore we can write it in te following form; d d d 3 d 4 S D = d d d 4 d 3 d 3 d 4 d d, 9 d 4 d 3 d d were d =, d =, d 3 =, and d 4 =. Te stiffness matri is given by te following equation. β + d d β 3 + d 3 β 3 + d 4 S = S D + S C = β + d d d 4 β 3 + d 3 β + d 3 d 4 β 3 + d d. β + d 4 d 3 β 3 + d β 3 + d Te vector b is determined as, [ [ [ b3 S3, S = 3, u, b 4 S 4, S 4, u [ b3 = b 4 [ β + d 3 d 4 β + d 4 d 3 [ 4 = [ β + d 3 + d 4. β + d 3 + d 4

27 Now unknown values u 3 and u 4 are determined as, [ [ [ [ u3 S3,3 S = 3,4 b3 + u 4 S 4,3 S 4,4 b 4, 3 were te last vector on te rigt and side comes from natural boundary conditions, [ u3 = [ [ β3 + d d β + d 3 + d 4 +, 4 u 4 s β 3 d β 3 + d β + d 3 + d 4 + were s = β 3 + d β 3 + d d β 3 + d. [ u3 = [ β3 + d β + d 3 + d 4 + d β + d 3 + d 4 + u 4 s β 3 d β + d 3 + d β 3 + d β + d 3 + d Finally we ceck for te variation te in -direction; u = u 4 u 3 = s [β β 3 + β d β 3 β + d 3 + d 4 +, 6 as, s d d and u. It is clear from equation 6 tat we can not ave constant numerical solution in -direction wit given conditions. Tis effect can be reduced by using a finer mes. Furter more, we consider only te diffusion art and set v y =. Tat makes β i = β i = 6 v yc i. From equation 6 we ave, i.e., we ave no error in te -direction. u 4 u 3 =, 7 6. FEM Model of Darcy Flow in Porous Medium Te system of non-linear equations is reconsidered ere in two dimensions. ρ +.ρ v =. D T + q, t, 8 t ρ t + ρ v =, 9 v, t = λ P. P, t.v = RT, t,, t = ct, t. Initial and boundary conditions are also known. We aly te same rocedure as given in te section for system of equations in one dimension. Te above system can be reduced to te following two equations in two variables witout a source term, ρ t λρ. =. D, 3 ρ t λρ. λ. ρ =. 4 We take te following initial and boundary conditions,, t = = y ρ, t = = y 5

28 at Γ, = y, at Γ,, at Γ 3, y at Γ 4. { ρ = at Γ, y at Γ 4. Were Γ, and Γ are inflow boundaries. Te time discretization of equation 3 and 4 is done as follows semi-imlicit sceme, ρ + λρ. + =. D +, 5 ρ + ρ λρ +. λ. ρ + =. 6 We ave used Standard Galerkin Aroimation for bot equations. Te element stiffness matri for equation 5 is comuted as follows te diffusion art S D e is calculated in a revious section, 3 3 S ei,j = λ ρ k φ k φj l φl. φ i dω + S D ei,j, m =,..., n, S ei,j e m k= = λ e m S ei,j l= 3 ρ k φ k k= = λ 6 ρ i Te element mass matri is given by, 3 M ei,j = 3 l b jb l + c j c l φ i dω + S D ei,j. l= 3 l b jb l + c j c l + S D ei,j. l= e m k= ρ k φ kφ j φ i dω = 6 ρ i δ ij. 7 Similarly element matrices for equation 6 are given by, S ρei,j = λ 3 l 6 b jb l + c j c l, 8 l= M ρei,j = 6 δ ij. 9 Te comutation of b is elained in te section for te diffusion roblem and te matri equations ave te same notation as given for one-dimensional case, i.e., + = M + S M + b, ρ + = M ρ + Sρ Mρ ρ + b ρ. Te numerical solution of and ρ is given in Figure 8. In Figure 9 te function ρ is given at different iterations. We observe tat it reaces a steady-state. Tis was eected because te boundary values are indeendent of t. Te matri equations for te fully imlicit sceme are given by, + M new = + + S + ρ + new = M ρ + S + ρ M + Te numerical result in tis case is given in Figure. 6 + M ρ ρ + b + ρ b +.,

29 .8.6 =.5 ρ y.6.4. y Figure 8. Te Numerical Solution of and ρ wit Semi-Imlicit Sceme, D =, =., =..5 = ρ =5.5 ρ.5 y.5.5 y.5 = ρ = ρ y y.5 Figure 9. Te Numerical Solution of ρ wit Semi-Imlicit Sceme at different iterations. = ρ y y.5 Figure. Te Numerical Solution of and ρ wit Fully Imlicit Sceme, D =, =., =. 7. Conclusions In tis reort we comuted numerical solutions starting wit steady-state linear roblem in one dimension and ending wit a system of couled non-linear equations 7

30 in two dimensions. We observed a good agreement in numerical and analytical results were te comarison was ossible to make. Initial and boundary conditions effect te accuracy and stability of te solution, esecially wen tey are discontinuous or te gradient is large. For non-linear roblems, te Picard metod worked reasonably well. For furter work, few of te equations in te system we treated, sould be relaced wit more realistic modeling. Te non-trivial roblem domains sould also be considered. Te accuracy and te stability issues are also tere for te said system. Furtermore, te said system sould be discretized in conservative form so tat numerical results sould reflect te ysical laws, realistically. References [ Klaus A. Hoffmann, Comutational Fluid Dynamics, Vol I, 4e, EES, Wicita, USA, [ J. van Kan, A Segal, F. Vermolen, Numerical Metods in Scientific Comuting, VSSD, 4 [3 Randall J. Leveque, Finite Volume Metods for Hyerbolic Problems, Cambridge University Press, USA,. [4 J.N. Reddy, An Introduction to te Finite Element Metod, e, McGraw-Hill,