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using recovery methods in localization problems?

Submitted by djeehand on

Hi,

Stress recovery methods have been proposed to enhance the prediction of discontinuous fields in finite element (stress field in particular). Some popular available methods include those proposed by prof. Zienkiewicz et al., i.e. SPR or REP methods. Now, I was wondering how accurate/suitable those methods are when dealing with a localization problem? In such a problem, there is a high contrast in the stress and strain quantities near the localization zone, so I'm wondering if a recovery patch does not somehow smooth out the overall solution? I'm sure there might be some references of interest, but I couldn't find them in websites such as science direct.

Thanks for pointing me in the rigth direction.

Cheers

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Computational Mechanics Forum
Biswajit Banerjee

The question was indeed unclear.  What type of material are you modeling?  Why do you need stress recovery?

Stress recovery techniques are useful if you have purely elastic materials and no history dependent variables.   You cannot get localization in the material if it is strongly elliptic  which is what purely elastic materials often are.  If there is softening then some internal damage variable is usually involved that depends on the history. 

If you want to refine adaptively around a region of localization then the stress recovery technique will not give you  accurate values of stress at arbitrary points because the internal variables at those points cannot be calculated accurately.

-- Biswajit 

Thu, 11/08/2007 - 18:18 Permalink
djeehand

Hi,

Thanks for your answer. I want to model plastic materials with strain softening. Now, the formulation is based on the material point method, with a "geometrical" mesh composed of quadrilateral elements (linear interpolation in velocity + constant pressure). With this formulation, the pressure is only accurate where it is calculated, i.e. at the centre of each element. I'd like to use this raw pressure in conjunction with recovery methods to get a better estimation of the stress field (which is used in the plastic criterion). But as soon as localization develops within the material, I'm wondering if it's feasible to use such recovery methods since the fields I want to recover are then strongly  discontinuous.

I don't know if that's the same idea you have in mind when you say that the internal variables cannot be accurately calculted at those points?

Cheers 

Thu, 11/08/2007 - 22:49 Permalink
Biswajit Banerjee

The Zhu-Zienkiewicz stress recovery method assumes that the stress field is smooth and continuous within a "patch" of elements surrounding a node.  The identification of superconvergent points, and hence the accuracy of the extrapolation, depends strongly upon the degree of smoothness assumed.  If there is a elastic-plastic boundary in such a patch then an additional complication arises.  The simplest phenomenological plasticity models assume that there is an internal variable (often called the "plastic strain") that depends upon the history of the deformation.  To calculate the stress at superconvergent points for such a model, one needs to know the plastic strain at the supercovergent points.  And even if the stress is calculated accurately at superconvergent points, one needs to make sure that the actual elastic-plastic stress field and the assumed smooth stress field are reasonably "close" to each other - say in a least squares sense.   Making sure of that "closeness" is difficult.

Now, let us consider the case where the deformation is localized in a thin region that intersects a patch of elements.  The problem is exacerbated because closeness of the solution to the assumed field (in L2) is no longer very meaningful and some other closeness norm is needed (say L_infinity).  You will then find that unless the assumed field is essentially the same as the actual field, the results will be off by quite a bit.  One solution is to refine until the actual field in each patch is close to a simple assumed field.  In that case the plastic strain has to be interpolated between known points/values - which will lead to inaccuracies and numerical diffusion.

In short, the ZZ stress recovery method is not the right technique for problems that involve localization.

Regarding MPM and the interpolation of pressure, I think that you can just use a constant pressure in each grid cell (because that is what you are assuming) when you calculate the stresses at the particles inside that cell.  I've tried a least squares fit for the pressure and a constant pressure and found that there is not much of a difference for the Taylor impact problem.  Also make sure that the domain of integration that you use for your MPM calculation is accurate - there are some subtle issues involved there. 

-- Biswajit 

Mon, 11/12/2007 - 17:47 Permalink
Biswajit Banerjee

What I meant was that in MPM the grid cell does not necessarily represent the physical domain over which you do your integration.  If the particles are represented as delta distributions then that is not so much of an issue.  However, we know that a combination of linear grid basis functions and delta distributions on particles leads to a unstable numerical scheme.  Scott Bardenhagen tried to address that issue using GIMP where the delta distributions are replaced by a piecewise constant/linear etc. basis.  This introduces additional complications regarding the domain over which to integrate, especially near the boundaries.

More recent developments show that you can continue to use delta distributions on particles as long as you use at least cubic splines on the grid.

-- Biswajit 

Sun, 11/18/2007 - 18:39 Permalink