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2nd spatial derivatives from isoparameteric elements
Based on my limited knowledge of FEM, I find it is difficult to use isoparametric elements for any problem whose governing
functional involve the second (or higher) partial derivatives of the unknown
functions. I hope some computational mechanicians can help me on this. Currently, I have a formulation derived from micropolar elasticity which has a functional involving second partial derivatives of the displacements. Thanks a lot for any of your suggestions.
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Re: Higher-order isoparametric elements
C0 higher-order isoparametric elements can be created quite easily using products of 1-D Lagrange polynomials. Are you running into problems while calculating the Jacobian? Or do you require higher than C0 continuity at the nodes?
Smoother isoparametric basis functions can be quite difficult to devise in more than 1-D. As an alternative, you can use normalized B-splines (with some tweaking) to achieve C1 continuity, or use some recent ideas such as MaxEnt (though I'm not sure how difficult it is to get C1 with MaxEnt. Sukumar?).
-- Biswajit
Smooth basis functions
Biswajit,
With MaxEnt, one can use a smooth weight (prior) to get C1 or smoother approximations. For linearly reproducing
basis functions, the code I posted here
w'd suffice, where expressions for 2nd order derivatives of the basis
functions are provided. Marino and Ortiz used Gaussian priors to obtain
smooth basis functions, and they have also recently developed smooth,
quadratically complete MaxEnt basis functions. Here's a discussion on C1 NEM shape functions, which are like an extension of 1-D Hermites to 2-D.
2D elements with higher derivatives
Biswajit and Sukumar,
Thanks a lot for your reply! I am dealing with 2D
elements. I do need C1 continuity at the nodes, which corresponds to
the rotation vector in the couple stress theory, which is equal to the
curl of the displacement. I have one code written in traditional finite element method including first and second order triangular elements and first and second order quad elements. I want to minimize the effort to extend it to the current problem. Sukumar, some terms such as reproducing basis functions in your reply I don't understand. Please forgive me that my computational mechanics knowledge is limited only to traditional FEM textbooks.
Wenbin
You need not introduce 2nd
You need not introduce 2nd derivatives in micropolar finite element analysis because microrotation is a variable independent upon displacement. For details, see, e.g.
Malcolm D.J.: Int. J.Engng.Sci., 20(1982), 1111~1124;
Nakamura et al: Int.J.Engng.Sci. 22(1984), 319-330
or Kennedy T.C. & Kim JB. Computer & Structures, 45(1992), 53~60;
But for couple stress problem, which is a specific case of micropolar, you must take care of numerical locking.
Xi, Thanks a lot for
Xi,
Thanks a lot for pointing out the difference between micropolar and couple stress. I am learning. Then can you tell me how to deal with or avoid numerical locking in couple stress problem?
Wenbin
Something like selective
Something like selective reduced integration method (e.g. Shu & Fleck: Int. J. Solids Structure, 32(1998), 1363-1383) or hybrid element( Herrmann LR in Hybrid and Mixed FEM, 1983, 1-17)