## You are here

# Banding in FEM

I know the following: In linear FE analysis with linear constitutive law, say CST, there will not be any discontinuity across the adjacent (or neighboring) elements, for any of the fields---displacements, strains or stresses. But I do have certain questions that are not very well addressed in the introductory FEM texts:

(i) Assuming a displacement-based formulation, under what conditions would you expect discontinuities (or inter-element banding) to possibly appear in: (a) stresses? (b) strains? (c) displacements?

(ii) Are there any guidelines whereby the banding can be expected *not* to occur in the above three fields, if the constitutive law is nonlinear? What if a higher order analysis is being conducted? (Here, I can't be sure if I have not made a mistake in a recent post in the point no. (iv) of my comment at: node/1957#comment-5377)

(iii) What precisely are the reasons to avoid a force-based formulation?

-----

It seems like the existence of banding does not depend on the form of the law---linear or nonlinear---so long as it's a monotonically increasing function and the point of fracture is not reached.

What I am looking for is a systematic account and well thought out replies. Thanks in advance.

## re: banding in FEM

Ajit,

Unless there is some miunderstanding in terminology here, your first sentence is incorrect. Classical finite element shape functions only possess C^0 continuity. Thus, if they are used to approximate a primary field (such as a displacement), there will not be any discontinuities in that field using a standard mesh. However, the gradient of this field will be discontinuous across element edges. So we should expect jumps in the strain field between elements, as well as the stress. This is completely independent of the constitutive law. Many commercial finite element codes do hide this feature by smoothing the strain and stress fields, making visualization simpler.

## Re: Banding

Hi John,

Your reply came in even as I was editing a bit quite some time *after* posting... I realized on my own that I was not being accurate about the constitutive law and all.

Thanks for correcting me.

Also, yes, I am talking of the classical element shape functions alone. Constant strain elements of course, won't have banding, quadrilaterals would. So would higher order elements. The question is, could you please recommend a book or paper wherein they focus on the aspect of banding as such?

## re: banding (again)

Ajit,

Hmm. I think there must still be some miscommunication here. Let me try to be as clear as possible.

With the CST, the strain in each element is constant. However, between adjacent elements, we should expect a jump (i.e. discontinuity) in the strain.

With bilinear quadrilaterals, the strain has a linear component, so there is some variation over the element interior. Between adjacent elements, we still expect discontinuities in the strain.

Now, perhaps by "banding" you refer to the case of a discontinuity within the element itself? To my knowledge, this issue is not discussed in any book. I'll look around and see what I can find in the literature, but I doubt there is much.

## Re: banding (again)

Hi John,

By banding, I mean the following. Consider a 2D domain. Without applying any smoothing, plot contour lines (of suitable increments) for the field in question. Now, consider the contour lines representing the same numerical value in each of the two elements. If these two contour lines happen to meet at the same point on the edge common to the elements in question, then there is no banding. If the contour lines end at *different* points on that common edge, then there is banding.

I prefer to call it banding (rather than discontinuity) because I better like a visual description like that. If you paint the strips between the contour lines in alternative black and white, like zebra strips, then the mismatch is very easy to spot. (The feature is very easy to implement in software.)

So, I was not referring to the banding within the element.

Now, after being corrected, I make the following statements. Please see if I now got it right...

-- CST will have no banding in displacement, but banding will be present (in the general case) in strain (and so stress) fields. (I was making a mistake about banding being absent in the strain field even in the general case here.)

-- LST will in general have banding in all the three fields: displacement as well as strain and stress fields. Also, the bilinear quadrangle element and the higher elements.

Also, what are your thoughts about that other question I raised: If banding in stress fields is so prevalent, why not think of using the formulation in which force is the primary unknown? What reasons---apart from tradition---go against it?

Another point. If the displacement fields also suffer banding in all the higher elements starting with the LST element, why do people set up such a huge store about FEM satisfying the compatibility criterion automatically? After all, it's compatibility only within an element....

But anyways, more important to me is the point about force as the primary unknown... Any thoughts?

Also, how does XFEM fare w.r.t. considerations such as these? Other "mesh-free" methods? What about incompatibility from point to point? Thanks in advance.

## banding in general

Ajit,

I'm not sure what you mean by the LST - this would be a triangular element with quadratic displacements, and thus linear strains? Such an element will not exhibit banding in the displacement field. Neither will the bilinear quad nor any other higher-order element. Only an incompatible element will exhibit banding in the displacement field.

Why don't people use a force-based method? Probably because most don't see banding as a problem. The only instance it really presents an issue (in my opinion) is when one needs to remesh/remap state variables between two different meshes. But it's not the case that they aren't used. Methods that yield smooth stress fields are used for error-estimation, for example, and some researchers employ mixed formulations in which both stress and displacement are primary fields. I don't advocate the latter due to the considerable increase in computational cost.

I believe my correction addresses your point about compatibility. Displacement fields don't band for higher order elements.

X-FEM is just like FEM on this issue. Most mesh-free methods use very smooth basis functions, and as such will not exhibit any kind of banding.

## Second thoughts on banding

Actually, for the CST element, the very idea of banding (in the sense defined in my above post) is inapplicable.

There will be a *discontinuity* in the strain field from triangle to triangle. However, since each triangle carries the same value of strain at all points within itself, there are no contour lines to think of *within* an element. As such, the issue of having a mismatch in their terminating points at the common edge simply does not arise.

Further, for modeling with the CST element, what you often do is the following. Take a common node shared by, say, three elements. Take the average of the strain values of these elements and assign it to the common nodal point. Having indirectly *derived* the nodal strain values in this way (not averaged but derived), we can interpolate the so-derived nodal values of strains to the points lying within the triangles.

Now, with this procedure, there are variations in strain values within the triangle, and so, we can at least think of the existence of contour lines within the element. Hence, the idea of banding now becomes applicable. However, since the interpolations are linear, there still are no mismatches along the common edges. Hence, banding never shows up in CST.

Please note, the above procedure is actually different from the smoothing algorithms followed for the higher elements. In the above procedure, we are not so much concerned with smoothening as we are concerned with the ability to at all derive contours. I have been implicitly referring to this procedure in my statement about banding.

-----

Of course, let me emphasize, John's description in terms of discontinuity is, fundamentally and rigorously speaking, the correct description.

The above procedure imparts contours to CST analysis artificially. So, even if it isn't an arbitrary smoothing, the description still isn't just as rigorous.

Describing other elements in terms of banding, however, does remains a rigorously valid idea.

-----

As I noted, banding (in the sense I defined in the post above) in strains comes up in all the other element types anyways. Also in displacements themselves!! Too bad!!!

## Re: Banding and plotting smooth stress fields

I believe that commerical softwares use the superconvergent patch recovery technique to compute nodal stresses and then do the isosurfacing based on those. The original papers are the first two in the following list generated by Google scholar. They are quite readable and worth taking a look at.

Google search: Superconvergent patch recovery

Biswajit

## Displacement discontinuity?

Hi Ajit, I don't see how banding(I prefer to say discontinuity) can occur in the displacement field if you have C0 continuity in the shape functions. Isn't the discontinuity in the strain field simply because the derivative of the displacement field being not necessarily continuous unless you take the derivative it self as an additional primary variable? Also, how does the choice of your element affect strain discontinuity? (I understood "banding (in the sense I defined in the post above) in strains comes up in all the other element types anyways" implies that)

## Re: Displacement discontinuity?

Hi Temesgen,

Yes, you are right. The C0 continuity will of course ensure absence of banding in the displacement field. The point is, you don't always get even the C0 continuity when you cross element edges.

You are also right about discontinuities in the strain arising out of derivatives---the C1 and C2 continuities aren't guarunteed by C0...

How does the element choice affect discontinuity.... I guess there is no easy way to answer this question, except in reference to the relevant maths. But I will give it a try.

Vaguely speaking, try visualizing keeping a 2D element (say a triangle) on the xy plane and plot the scalar field for one of the displacement component (i.e. u or v) along the z axis. In the general case, the triangle would deform in such a way that its edges become curves in this kind of a plot---not straight lines. Now, given a pair of points, you can pass only one straight line through them, but when it comes to curves, you can pass as many of them as you like through them---an infinite number of them. So, as soon as you relax the requirement that the deformation of the element ought to be such that it keeps the straight edges straight, you no longer have a unique edge between the two elements---each element will curve differently (depending on how the potential function minimized within its own reference domain). Translated, this means that there is no C0 continuity in the displacement field across the two elements.

## re: displacement discontinuity

Ajit,

The situation you've described does not occur with elements based on compatible displacement fields. For a quadratic triangle, for example, only a quadratic curving of the edge is perimitted in the deformed configuration. The curve is uniquely determined by the same set of common basis functions (and their displacement degrees of freedom) shared by each of the elements sharing the edge. This can be generalized to higher order elements, and so on.

## Re: Displacement discontinuity---Got It!

Hi John,

Yes, that's right. Finally, I got it. In terms of the description above, what happens is this: The way the element edge can curve (instead of being straight) is, say, if it is a polynomial (to take an example). The specific shape assumed by a particular form of polynomial is completely determined by its coefficients. Now, the thing is the following: The FEM formulation enforces identity of the coefficients for adjacent elements. (That's what assembly of local matrices to the global matrix is all about.) Therefore, the adjacent elements curve identically at their common edge---it's one and the same curve they come to share. So, C0 is automatically ensured for displacements. However, since the "curvature" in the direction normal to the edge is not necessarily the same. Hence, C1 and C2 may not be ensured.

I was somehow not able to keep the idea that the FEM formulation would enforce identity of the coefficients in complete focus---somehow, I would refer to only the end-nodes and not the nodes in the middle of the edge.

-----

Actually, this means I also made a mistake in formulating one of the questions in a simple quiz (mentioned on my personal Web site) too. Of course, there, I was still only designing the course---I had not yet even begun working on it. I would have surely gotten this issue resolved from someone (the way I raised this question here) before going out to teach FEM to someone. ... Anyway, now I will go and change the quiz. (Actually, it was months back, while designing that same quiz that I started thinking about this issue.... Somehow I forgot about it before posting the quiz on my Web site!!)

-----

Thanks are due to all of you: John, Temesgen, Biswajit, Ramdas, for responding and staying with me until I "saw it"!

-----

Before closing, just a word about where I got this word "banding" from. I actually checked it and found that I had picked it up quite some time back from Cook, Malkus, Plesha's book---pp. 581 in their III edition. They quote the word "stress-band" as coming from: T. Sussman and K. J. Bathe, "Studies of finite element procedures---stress band plots and the evaluation of finite element meshes," Engineering Computations, Vol. 3, No. 3, 1986, pp. 178--191. ... Just thought of jotting this down because everyone else other than me here was calling it a discontinuity and not banding... Cook et al.'s book also has an accompanying illustration of banding. It was that particular visualization which had stayed in my mind...

Thanks again and bye for now...

Ajit

## R. Chennamsetti, Scientist,

R. Chennamsetti, Scientist, R&D Engineers, India

Commercial softwares find out the stresses/strains at Gauss points, interpolate these values to nodal values using shape function obtained by taking Gauss points as co-ordinates. If a node is shared by 'n' elements, then the stress computed from each element at that node is summed out and calculates the average for all six components. Then we get one representative value of each stress component at the node. Now one can compute the required stresses like von-Mises, Tresca etc based on the theories of failure. In this case, the principal stresses are calculated from averaged stress components.

In the second approach, in an element, interpolate all six components to nodal points, then calculate the principal stress components. We get three principal stresses from each element shared by that node. Now take the average of these principal stresses, get a representative single component for each principal stress component, then compute the required stresses like von-Mises, Tresca etc , from averaged principal stresses, based on the theories of failure.

Thus in commercial s/ws we get nice contonurs of stresses and strains.

As mentioned by Prof. Dolbow, in C0 continuity elements, we get only dependent variable like displacement is continuous, but not the stress/strain.

With regards.

## stiffness matrix calculation of incompatible elements

I am a student of M. Tech. 2nd year. My project topic for dissertation is "Numerical & Experimental Estimation of Fracture Toughness of Some Materials using Compact Tension Specimen". In this dissertation I also wanted to develop a code which directly gives me the values of stress intensity factor just like ansys.

Actually I have develop an matlab programm (using linear quadrilateral element) which gives me the value of displacement at the cracktip using that i would be finding the displacement extrapolation method i would be calculating stress intensity factors

But the major hindrance in my matlab code is that when i am calculating the stresses at the crack tip there is large deviation in cracktip stresses shown by matlab code and ansys. Looking at the ansys theory reference it was found that apart from four shape function i.e, N1, N2, N3 and N4 they are using two more shape function N5 and N6 whose values are (1-s^2) and (1-t^2) s for x-direction and t for y-direction respectively.

Also such type of shape function is described in R. D. Cook, Bathe and Zienkiewics books which they are calling as non-conformal elements which are used to remove the shear lock effect of linear quadrilateral elements and having co-efficient as a1 a2 for x-direction and a3 and a4 for y-direction. But nothing has been stated about the values of ai (i varies from 1 to 4).

Hence Sir, I would be most thankful to you if you could give me guidelines regarding how to solve such type of FEM problem having non-conformal elements or u could give me reference where

Problems are solved using Non-conformal elementsI would be most thankful to youRegards

Zoeb Kaizar Lakdawala

M. Tech (Industrial Process Equipment Design)

S. V. National Institute of Technology, Surat

Mobile : 09904288552