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September 16, 2002, 13:29 
A (very, very) stupid FEM Question

#1 
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Yes, I know, it is a very stupid question, but I did not still find the answer to my problem (and I prefer be a public stupid rather than stay with the doubt ....)
Here is the problem: It is about the Q2 Finite Element and Affine Mappings. I'm writing a FEM code with the Q2Q1 Element (Quadratic velocity and bilinear pressure), and I want to use an affine mapping from [1,1]^2 to each rectangle K (and so, the mapping [1,1]**2>K is Affine. So, If I define the REFERENCE basis velocity functions fi_1....fi_9 for [1,1]**2, and I want to utilize these functions to calculate the integrals over each other element in an expression which have a derivative, for instance, Integral (fi * d/dy(fi)) I know that in the corresponding transformate integral over [1;1] the (constant) Jacobian has to appear, But do I have to put also some (constant) scaling for the derivative? (something like 1/hx or 1/hy depending the direction of the derivative). The logics says to me "yes", but if I put this "scaling" in the derivatives (and keeping always the Jacobian), my iteration diverges, but if I use only the jacobian, and perform the integrals without this scaling, the iteration converges. So, the questions are (for the AFFINE case AND Q2 QUADRILATERALS!): 1 It is necessary to compute explicitely the basis functions in each element in order to calculate the terms for the variational formulation? 2 It is necessary to calculate another basis functions that the [1,1]**2 basis functions? Remark: I perform Gauss quadrature, so my interest on the reference element. 3 There is anybody that have a Fortran subroutine to calculate these functions or at most to see how calculate the integral in every element? (I have the nodes and connectivity array for each element) Thanks in advance, and sorry, but it is my first experience in the Finite Element Utilisation.... PS: Yes, I have read FEM tutorials, but almost all deal with the isoparametric formulation, not sub parametric. 

September 20, 2002, 10:12 
Re: A (very, very) stupid FEM Question

#2 
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Since no one else has responded, Carlos, I will share my limited experience on the question. Assuming you have the correct form for the derivatives, I would look for a coding error. In principle, the scale factors are necessary, but sometime there is so much noise in other approximations that you can't see the effect.
I have generalized the Q1 and Q2 to complete linear and quadratic elements which have constant divergence. This constant divergence vanishes pointwise on an element provided the net flow into the element is zero. The elements are sufficiently continuous that the divergence vanishes on the element boundaries as well. Examination of the elements shows that the element geometry can be factored out into pre and postmultiplying matrices. These diagonal matrices can be recognized in terms of Jacobian matrices of the element transformation. This is the basis for generalization to affine elements. The premultiplying matrix is J/det, where J is the matrix of the transformation (constant for affine transforms) and det is the determinate of J. I have been told that this is called the "Pioli transformation" but I have never seen a reference. The postmultiplying matrix is the inverse of this, evaluated at the node. Using a symbolic math program (such as Maple), or by hand if you are young enough, it can be shown that this form identically interpolates all linear or quadratic divergencefree velocities under affine transformations. Use the chain rule to get derivatives of the velocities. The coordinate derivatives can be evaluated from the components of J. These derivatives are also exact for affine transforms. Matrix elements are evaluated in terms of integrals over the reference square and products of J and J^1 and components thereof. If you generate a mesh with coordinates and J at each node, evaluation of the matrix elements is very easy. On more general structured meshes, even curved elements are often approximately affine. One could use the exact Jacobians at the Gauss points, but the affine approximation works quite well. You would have to investigate for yourself to see how these comments apply to Q1 and Q2 elements. 

September 21, 2002, 09:31 
Re: A (very, very) stupid FEM Question

#3 
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Hi, Jonas,
Thank you very much for your smart answer. I knew that it will be hard to have an answer to these questions, but now with your help and some others FEM books I begin to understand. I encouraged to the other people to answer the questions which are posted, even though they you look too stupid. Anybody is born by knowing everything! And this is the idea of a forum, as far as I know... Thank you again, Jonas. Carlos. 

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