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December 11, 2008, 13:53 
FEM num integration: quadrature rules for a cube

#1 
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Hi all, I am looking for a web site where I can find some high order quadrature rules (i.e. integration formula) for a cube, which are exact up to order 7. I am using Q2 cubic elements in my FEM code.
I don't want to use any 3D generalization of a 1D nintegration points (IP) Gauss formula because the number of IP varies like n.n.n, which is huge. I know some cheaper (less IP) integration formula exist but did not manage to find them on the web. Seems to be quite tough to find a web site that gather these formula. Thanks for your help, Anthony 

December 11, 2008, 14:51 
Re: FEM num integration: quadrature rules for a cu

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Encyclopaedia of Cubature Formulas:
http://www.cs.kuleuven.ac.be/~nines/...h/ecf/ecf.html I don't know what you mean by Q2 cubic elements. Q2 elements usually mean tensor product quadratic which means the maximum order is 6 and you end up not beating the tensor product rule. If you are using a truncated space then the cubature rules above will help. Note that if the basis and quadrature rule are tensor products, you can exploit the tensor products for evaluation, never forming the <code>Q\times P</code> basis and derivative matrices. This operation is sufficiently fast that I don't even form the Jacobian for Q2 and higher order, I just form a Q1 preconditioning matrix. This uses half the memory and the linear solver (preconditioned with ML's parallel algebraic multigrid) converges in half the time of an assembled Q2 matrix, but I get arbitrary spectral order (Q2Q8, then the tensor product operation breaks out of L1 cache so further prefinement is less efficient). If you're going to bother assembling the matrix, by all means use truncated spaces and the cubature formulas, but tensor products are nice too. 

December 12, 2008, 06:07 
Re: FEM num integration: quadrature rules for a cu

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A Q2 cubic element is a finite element the shape of which is a cube with 27 nodes (8 corners + 12 middles of edges + 6 centers of face + 1 center of the cube = 27). The basis functions of each node are quadratic (2d order polynomial) i.e. Q2.
What I am looking for is a Gauss or NewtonCotes like integration formula to compute the sum of a polynomial function of order 6 or 7 on the basic cube [1:1] x [1:1] x [1:1] or alternatively [0:1] x [0:1] x [0:1]. With this type of element, the mass matrix terms are polynomes of order 6 (w_i*w_j*det J = 2+2+2 = 6 ), that's why to integrate these terms exactly, I need a quadrature rule or order 6 or 7. Hope it is clearer right now, sorry to have been cryptic. Anthony 

December 13, 2008, 15:48 
Re: FEM num integration: quadrature rules for a cu

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I thought cubic possibly referred to order 3 polynomials which didn't make sense. I don't think you can beat the tensor product quadrature rule here, just use the 3*3*3 Gauss rule. If you use a truncated space (like corners + edges) then there are better rules (see the link in my last post).
Note that there <code>f = (1x^2) (1y^2) (1z^2)</code> is the center basis function so your mass matrix needs to integrate <code>f*f*det(J)</code> which has order at least 12 (depending on the element mapping type). There is a big difference between getting all polynomials of order 12 (which requires many points) and just tensor products of quartics (which the 27point rule will do). As my last post indicated, assembling a matrix based on Q2 or higher elements is usually counterproductive. 

December 16, 2008, 08:36 
Re: FEM num integration: quadrature rules for a cu

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Thx Jed, Anthony


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