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From: John Peterson <peterson@cf...>  20070716 22:33:09

Anurag Singh writes: > Hi, > > As per John's suggestion, I'm emailing it to the whole users list for some > very simple questions on example 13: > > > > . We are solving the gov. equations for a lid driven cavity in 2D: > > rho*(v_i,0 + v_i,j*v_j) + p_i  mu*v_i_jj = 0 > > v_ii = 0 > > > > The FEM formulation of the above equations in example 13 is not clear to me. > Can anyone help? The difference here may be that the problem is nonlinear, and you have solved only linear problems in the past? We have a general nonlinear problem F(U)=0 where U is a vector of FE coefficients. We use Newton's Method to arrive at the linearized equations J(U_{k})(U_{k+1}  U_{k}) = F(U_{k}) J(U) is the Jacobian matrix of partial derivatives, U_{k} is Newton iterate k. Search for multivariable Newton's Method if this looks foreign to you. The nonstandard thing which is done in ex13 is to move part of the LHS of Newton's equation to the RHS as: J(U_{k}) U_{k+1} = F(U_{k}) + J(U_{k}) U_{k} This might be why the righthand side does not look familiar to you. The linearized equations are solved multiple times per timestep until the solution converges. The time discretization is a thetascheme. Theta=1 is the same thing as backward (implicit) Euler. John > > > Boundary conditions are implemented using penalty method. > > > > I couldn't figure out the exact formulation to figure how it's doing it. > > > > (I'm not very knowledgeable about FEM and trying to relate what I read in a > book e.g. for say linearized burger's equation v_i,0 + vbar_j*v_i,j  > nu*v_i,jj  f_i = 0 with generalized galerkin methods leads to the > formulation (A + dt/2*(B+K)*V^n+1 = [Adt/2*9B+K)]v^n + dt*(F + G)). (A is > mass matrix, K is stiffness matrix etc. > > Now this is basically in the form A*u =B . which is then solved for u. The > formulation is DST approximation. > > > > So basically how does the ex13 implementation translates to something like > above. > > The book I'm referring is Computational Fluid Dynamics by T J Chung. > > I'm pretty sure it's quite simple and those who have done it can quickly > scribble. I might be able to figure it out eventually as I get proficient in > FEM but it would save me a lot of time if you folks can help me here. > > > > . The above should hopefully also answer how we have a terms like > Kuu, Kup , Kuv etc. which are themselves matrices. > > . I wanted to compute Re of the flow but couldn't find the value of > "nu" used in the problem? (L & u is 1 but don't know nu, 1/300 ??) > > > > Any help would be greatly appreciated. Unless I understand all this I'm > finding it difficult to move on. > > > > Regards, > > Anurag > 
From: Anurag Singh <anurag@st...>  20070716 20:45:58

Hi, As per John's suggestion, I'm emailing it to the whole users list for some very simple questions on example 13: . We are solving the gov. equations for a lid driven cavity in 2D: rho*(v_i,0 + v_i,j*v_j) + p_i  mu*v_i_jj = 0 v_ii = 0 The FEM formulation of the above equations in example 13 is not clear to me. Can anyone help? Boundary conditions are implemented using penalty method. I couldn't figure out the exact formulation to figure how it's doing it. (I'm not very knowledgeable about FEM and trying to relate what I read in a book e.g. for say linearized burger's equation v_i,0 + vbar_j*v_i,j  nu*v_i,jj  f_i = 0 with generalized galerkin methods leads to the formulation (A + dt/2*(B+K)*V^n+1 = [Adt/2*9B+K)]v^n + dt*(F + G)). (A is mass matrix, K is stiffness matrix etc. Now this is basically in the form A*u =B . which is then solved for u. The formulation is DST approximation. So basically how does the ex13 implementation translates to something like above. The book I'm referring is Computational Fluid Dynamics by T J Chung. I'm pretty sure it's quite simple and those who have done it can quickly scribble. I might be able to figure it out eventually as I get proficient in FEM but it would save me a lot of time if you folks can help me here. . The above should hopefully also answer how we have a terms like Kuu, Kup , Kuv etc. which are themselves matrices. . I wanted to compute Re of the flow but couldn't find the value of "nu" used in the problem? (L & u is 1 but don't know nu, 1/300 ??) Any help would be greatly appreciated. Unless I understand all this I'm finding it difficult to move on. Regards, Anurag 
From: hallmark.com <xvd@pp...>  20070716 02:55:41

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