Post Processing in FEM
Post Processing in FEM: Least Squares Smoothing v/s Direct Evaluation at Nodes
I have a question or rather an observation that I would like you all to throw some light on. I wrote a 2-d FEM code for solving steady state and transient linear problems in thermal conduction and stress analysis. I am using Rayliegh-Ritz method with Iso-parametric formulation. My element library has 3-node linear triangle, 4-node bilinear quadrilateral and 6-node quadratic triangle. The code has been validated throughly. My questions pertain to the post processing for 4-node Quad and 6-node Tri elements. A small subprogram computes nodal heat fluxes/stress values. As you all know there are many different approaches to post processing. I used three different approaches, taken from variety of sources (online notes, books, thesis etc...). I tested the accuracy of post processing results for 2 thermal conduction cases in a 1X1 square plate and the classic stress concentration problem (Square plate with a centered circular hole). The analytical solution (series solution) was known for the Thermal Conduction cases. The analytical expressions for nodal heat fluxes was obtained by differentiating the analytical solution, so that I could compare the accuracy easily. Here's what I observed !!! Any Comments ??? Explanations ????
Approach 1: Evaluate nodal heat flux/stress using same number of Gauss points as the number of nodes in the element (i.e 4 Gauss pts for 4-node Quad and 6 Gauss Points for 6-node Tri). The values can then be interpolated to the element nodes. Finally the average nodal values are computed by area weighted averaging. (This approach is described in Online notes : Intro to FEM by Prof. Carlos Felippa,Aero Eng Dept, UC-Boulder)
Approach 2: Evaluate nodal heat flux/stress using optimal number of Gauss points. ( One point (psi,ita) =(0,0) for 4-node Quad and 4 Gauss points for 6-node Tri). Then use LEAST SQUARES smoothing and interpolate it to nodes. (This approach is from book Finite Element Analysis from Concepts to Applications by David Burnett.) According to the book it is more accurate that direct evaluation at the nodes. the book also recommends averaging as in Approach 1.
Approach 3: Evaluate nodal heat flux/stress directly using the natural coordinates of the nodes and later average out as in Approach 1. (I think this is in Reddy's book)
I use L-2 Norm to compute the error in post processing for thermal analysis.
SURPRISE !!!! Approach 3 gave me the best values for predicted maximum stress and nodal heat fluxes for both elements.
Approach 1 is only slightly less accurate than Approach 3
Approach 2 . The error increases by 8 - 10 % ????
OFCOURSE !! I am not ruling out bugs in the code. Still I would love to hear from you all about what technique you all use for post processing. Sorry for the long posting. I have to be as clear as possible.
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