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Topic: FEM: strain-displacement matrix for isoparametric tetrahedral elements
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Jörg

Posts: 3
Registered: 7/8/12
FEM: strain-displacement matrix for isoparametric tetrahedral elements
Posted: Jul 10, 2012 3:21 PM
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Hi Group,

again I'm dealing with the basics of the Finite Element Method. I found a very nice PDF on tetrahedral elements: [1]. I'm intersted in the "general" / "arbitrary" Iso-P Tetrahedron. But I have a little problem figuring out the derivatives of the shape functions for global coordinates.

[1] http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch17.d/AFEM.Ch17.pdf

At the top of page "17?12" they give the partial derivatives for a function F.
dF/dx = F_k * dN_k/dZeta_i * a_i/J

where "F_k" is the value of the function "F" at node "k". "N_k" is the shape function of node "k" and "Zeta_i" is the i-th local coordinate. "a_i" and "J" are also given.


My problem: I need the derivative of N_k with respect to the global coordinates. I like to assemble the strain-displacement matrix '[B]' of this values. Can I get this from the function above? My guess is for a given node:

dN/dx = (zeta_1 * a1 + zeta_2 * a2 + zeta_3 * a3 + zeta_4 * a4) / J
dN/dy = (zeta_1 * b1 + zeta_2 * b2 + zeta_3 * b3 + zeta_4 * b4) / J
dN/dz = (zeta_1 * c1 + zeta_2 * c2 + zeta_3 * c3 + zeta_4 * c4) / J

where zeta_i is the value of the function dN_k/dZeta_i at the current location (in my case: one of the integration points).

Is my guess correct?

Thanks in advance,
Jörg



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