Zienkiewicz-Zhu superconvergent patch recovery (SPR) of stress


I’m looking at different stress recovery methods and their implementation. Regarding the CalculiX implementation the manual cites the following:

[54] Merz, S., Anwendung des Zienkiewicz-Zhu-Fehlerabschatzers auf Triebwerksstrukturen. Diplomarbeit-Nr. 06/56, Hochschule Karlsruhe - Technik und Wirtschaft, Fakultat fur Maschinenbau (2006).

Does anyone know where I can get a copy of this?

Also, in Zienkiewicz & Zhu (1995), the authors show that the SPR method can also be used for linear elements despite the integration points not having the superconvergence property. Could this functionality be added to CalculiX for linear elements (e.g. C3D4, C3D8) and for two-dimensional elements (CPS3, CPS4, CPS6, CPS8 etc.)? Currently it appears that only 3D quadratic elements are supported.

Hi Ben,

try Otto Bernhardi at the Fachhochschule Karlsruhe (University of Applied Sciences in Karlsruhe). He was the supervisor and may have an electronic copy. I only have a paper copy.

In the meantime we are using another error estimator; the ZZ error estimator had the problem that it did not necessarily decrease when refining the mesh.

Best Greetings,


Zienkiewicz SPR is a good one. You can also check this document,

From Carlos Felippa, at Colorado University, in the U.S.
Remember that whatever is done at the nodes, it is not the true value. It is just an extrapolation: SPR, Using shape functions, averaging, etc… The only “true” values are those calculated at the integration points and are those that should be really used in constitutive modeling.
This being said, since it is for visualization purposes, I suggest you to check the technique of weight averages for a given node. Using or not the shape functions. Where the weight, or the contribution of each integration node to the nodal value is given depending on how close are these to the nodes. The closest, the hightest the weight, and the sum of all weights =1. See for instance the contour plot on the left (CCX) and on the right (using the technique that I just mentioned) for a stress contour plot.