Stress at integration points or at nodes ? 7

[opethian]

Think of that you have a 3-D model and you simulated some forces acting on it. And get a Stress contour ? it should have normally the most accurate results at integration points (or not)? but then you have 4 integration points so should we extrapolate the results to the nodes ? or for this element which value should we take ? how about the stress results at nodes arent they realistic ?

 

[johnhors]

Hi pja,

Which Cook book are you referring to, is it :-

"Finite Element Modeling for Stress Analysis"

or

"Concepts and Applications of Finite Element Analysis" ?

 

[pja]

Concepts and Applications of Finite Element Analysis

 

[prost]

Not to sound overly critical, but have you ever constructed by hand a 'finite element'? It appears not, since you have stated that stresses and strains are needed.

To illustrate, I think I don't lose any generality by describing how the finite element matrix is constructed in 1D, for rods. We are solving Ku=f; 'u' is the solution vector, the displacements, we are trying to compute. 'K' is the stiffness matrix. 'f' is the force vector, which represents how we have loaded our body. In 2D, those are point loads, and distributed stresses (which are specified, so they are known and don't have to be computed).
Assuming linear elasticity, stiffness matrix elements k(i,j) look like (AE/l), where A is the area of the beam, E is the Young's modulus, 'l' is the length of the rod.

Stresses and strains are then computed AFTER you compute the displacement vector 'u', and are therefore never needed for the finite element solution process until the post processing (you might need to compute stresses to compute Stress Intensity Factors, for instance).

Certainly these equations are derived from the equilibrium of stresses, and you are computing spatial derivatives to derive the stiffness matrix elements k(i,j), so you 'sort of' are computing quantities that look 'like' stresses. However, just because you are computing spatial derivatives, doesn't mean IMO that you are computing strains and stresses. Therefore, I still cannot see reasons why and no one has provided a reference as yet that shows that stresses at integration points are more accurate (as measured by their convergence rates) than stresses at the nodes. I'd love to hear the explanation and see a demonstration, though. There's a lot to be said for personal experience, that's for certain, but its hard to use that in say, an internal or external report, so that observation is of little use to many FE analysts such as myself, since it represents something akin to hearsay or anecdotal evidence.

 

[prost]

pja-thanks for the reference. I have only 3rd edition of Cook, Concepts and Applications of Finite Element Analysis, available. Could you possibly supply the Chapter and section number (and/or title) to help me zero in on the correct section, since almost undoubtedly the pages are different from 3rd to 4th edition. Tks

 

[prost]

Apparently others here think that there are other ways to compute stresses than the obvious one (using spatial derivatives of the solution vector), so I'll do some more research to see what others are doing. I can't imagine how you can use the stiffness matrix and the solution vector to compute stresses, but it seems reasonable to assume there might be a way to do so, so it's worth more effort to track that down.

 

[EdR]

prost:

To try and answer your basic question..."why are stresses more accurate at integration points than at nodes" consider the following....

Assuming that we are talking about isoparametric element formulations then most element stiffnesses are computed using Gaussian integration to compute the stiffness (K=sum(Bt . C . B * wt * vol))where B is strain/displacement matrix, C is stress/strain matrix, wt is Gaussian weighting factors (dependent on number of integration points used), and vol is the volume fraction (also associated with the number of integration points used). If you investigate the Gaussian integration method you find that the results of the integration are most accurate at certain points (the Gauss points) for a given number of integration points (in either 1-D, 2-D, or 3-D).

Thus the strains (and stresses) that are computed as eps=B . U and sig= C . eps at the Gauss points are more accurate (optimal) than those computed at other non-optimal points as the B matricies used in the stiffness formulation are then consistent with those used to compute strains...Indeed the B matricies at some points (not Gauss points) can give very bad results for some formulations while the matricies at the Gauss points give good results..In other words being consistent when computing strains at the same points as those used when forming the stiffness gives the best results.....

For elements using other formulations and/or integration methods the above might change but most numerical integration methods have optimal points at which the integrations should be performed.....

Hope this helped....

Ed.R.

 

[pja]

Prost,
Chapter: Isoparametric Elements
Section: Stress calculation

 

[prost]

After more research, I am still stumped. Though I do have a few papers whose authors make the same claim about 'stresses at the integration points being more accurate than at the edges' matter of factly. So I have some more digging to do to figure out why. One person I know uses ANSYS I think said that when you request stresses, you can ask for them only at the integration points, which naturally leads to extrapolations at the edges and nodes. Don't know why there is that restriction, at the integration points, because it's unnecessary IMO. Since stresses are so important to the post analysis, direct computation anywhere in the domain you are analyzing seems like minimum capability.

 

[btrueblood]

"Not to sound overly critical, but have you ever constructed by hand a 'finite element'? It appears not, since you have stated that stresses and strains are needed."

Yes, I said so, have the degree and thesis in Aeronautical Engineering to prove it, have done so multiple times in the course of my career for multiple different problems, and cited the reference used, although I also extended said reference in several instances, to include thick-wall shell elements, elements undergoing rapid dynamic loading, and elements undergoing rotation and associated body forces...amongst other things. The reference given discusses the formulation of stiffness matrices using the techniques I described earlier, the use of Gaussian integration and errors associated with it (which EdR also describes), and how the use of Gaussian integration allows you to make statements regarding the solution's error bounds (whereas other techniques, such as Galerkin method don't allow for such treatment of error bounds beyond the technique of refinement and iteration). And your assertion that I don't know what I'm talking about irks.

Your next statement:

"To illustrate, I think I don't lose any generality by describing how the finite element matrix is constructed in 1D, for rods. "

Is specious. Yes, you can use "strength of materials" to compute 1-D and perhaps even 2-D (beam bending) element problems, then extend those to 3-D for frames and trusses. You can certainly ignore the rest of the FEA textbook, if you've ever read one, which I am beginning to doubt.

To extend the FEA method to even a 2-D general plane stress element (having both shear and normal tractions applied at the boundaries, having any general shape within certain limits) and then prove that the element is robust (capable of solving the problem under all possible combinations of shape and load) requires that you solve for the stress and strain field within it, in order to arrive at the stiffness matrix formulation. AFAIK, there are no shortcuts to that process, and the extension from 2D plane strain elements to 3D general-purpose elements is made using the same procedure.

 

[btrueblood]

Pardon the error, the last mention of 2D elements should have been "plane stress", not plane strain. The statement is still technically correct, I believe, but not grammatically so.

 

[prost]

Apparently, we are all reading different finite element textbooks, each with its own set of messages that the authors mean to convey. Therefore, we each have a different knowledge basis for comparing what we think we know about things like FE and what others seem to know. For instance, I learned from Szabo and Babuska, "Finite Element Analysis". This book discusses primarily the p-version of the FE method, but spends a lot of time discussing error analysis in engineering computations. As far as I can tell, not a single passage in Szabo's book talks about superconvergence properties of stresses at the Gauss points.

BTW, I am intimately familiar with about 75% of the Bathe FE book, since I needed it for explanations of the nonlinear deformation coding in ADINA. I don't recall any such superconvergence property, but I admit I could just have forgotten.

I have also taken 2 classes from Szabo himself (he was one of my thesis advisors, so he supervised the research), and have a PhD in mechanical engineering (which means dissertation of course--"Large indentations of nonlinear viscoelastic materials, with an application to cell poking"), research in analyzing nonlinear deformations of viscoelastic materials. I wrote two research FE codes, in addition to writing material constitutive subroutines that interfaced with ADINA. Does that mean I think I know everything? Of course not. This makes me the opposite of that--I am confident in what I know, but I am acutely aware of my ignorance in other areas. I will check out the one reference cited here (can't get the other one at the library), and maybe even work out an example or two to check if the same superconvergence property holds with higher order (p>2) elements.

It's possible I have read this before, with a proof, about the superconvergence property, and perhaps I forgot what the explanation was. I am sorry if I offended you; the explanation for the superconvergence property isn't so obvious, and I'm like most engineers, I think--I don't automatically assume when someone makes a statement that seems so obvious to him/her, that the statement is still true.

One last thing--because you are using the principle of virtual work (which is expressed in integral form) to create you FE matrices, you are not strictly enforcing the equations of equilibrium (which are most often written with stresses), point by point. This is why the FE equations are called the 'weak form' as compared to the 'strong form' of the equations of equilibrium. Therefore just because are using the equations of equilibrium to formulate the FE matrices, doesn't mean that equilibrium occurs at every point (or for that matter, any point in particular) in your domain. This also leads me to the conclusion that my stresses aren't more accurate say at the Gauss points in particular compared to anywhere else just because I am computing spatial derivatives just at those Gauss points. Point wise equilibrium just isn't being enforced point by point, so why should anyone assume engineering computations any particular point are more accurate than at any other point?

 

[btrueblood]

Prost,


I tried to point out in an earlier post, that the idea that stresses are accurate only at integration points is strictly true only for the FEA model that was constructed, not for the "real world", and you can find different stresses for the same real world points by refining or modifying the FEA mesh. Which implies what I think you are stating. Or perhaps not.

The problem is worsened (and I'm not going to address it any further as you sound like the expert here) by the idea of nonlinear, viscoelastic materials, where you use (correct me if I'm mistaken, it's been 25 years) separation of variables to solve for the instantaneous elastic stress, then apply the Duhamel integral to solve for the decaying time history of the viscous part of the constitutive relations. The "stresses are correct at the integration points" may be applicable to the instantaneous/elastic FEA model solution (depends on whether you are using Gaussian integration or another method), but the time history part I think induces a whole 'nother set of error analysis ideas more applicable to those used in finite-difference (i.e. CFD) codes... and the combination of the two techniques in the same solution would make me throw up my hands.

 

[prost]

My question had nothing whatsoever to do with the difference between measured and calculated stresses. My question had to do with the general principle of assuming calculated stresses are more accurate at the integration points than they are at the nodal points. My assumption was that was a limitation of some FEA codes that use isoparametric elements, and in way, I was correct, as some FE codes allow (or recommend) stress computations only at the Gauss points. In another way, I was way off base, in that this statement has a sound foundation. In asking for references, I was looking for something more of a theoretical basis rather than just experience based reasons.

It turns out there is a strong theoretical basis for this claim, which appears to have been proved for a few element types, but not all. Starting with Barlow, "Optimal Stress Locations in Finite Element Models," Int. J. for Num. Meth. in Engrg, Vol 10, 1976 pp. 243-251; Barlow demonstrated with a few examples that show that evaluation of stresses is 'optimal' at the Gauss integration points:
1) for beam with cubic shape functions if the exact solution is a polynomial of order quartic 2) for 8 node isoparametric plane elements (what many of you call QUAD8, right?) if the exact solution is a cubic, and 3) for 20 node isoparametric solid elements if the exact solution is a cubic. Barlow did not specifically use the term "superconvergence." However, this paper is a very nice introduction to the way an engineer might try to show 'optimality' of a particular computation.

While those Barlow examples might seem to be too specific, in that we cannot assume a priori that the 'exact solutions' are polynomials, there are at least 3 theoretical treatments that demonstrate the validity of the "stresses are more accurate..." statement: Strang and Fix, An Analysis of the Finite Element Method, Zienkiewicz, The Finite Element Method (I have access to the late 80s edition) and Wahlbin, Superconvergence in Galerkin Finite Element Methods. The last one is interesting, but readable only if you have some idea what a L2 space is! Wahlbin uses a common technique--first guess that the convergence rates for a certain calculation are of a certain form, say the 'norm' raised to a power 'q', then prove that the power 'q' is less than 1, and therefore the computation is 'superconvergent'. (Proving that 'q' is some value is IMO the most significant contribution the mathematicians have made to the theoretical development of the finite element method.) In a review by DN Arnold, a couple of results are relevant to this discussion were made clear: in one dimensional problems, and continuous piecewise quadratic elements, the finite element solution (that is, the displacements) are superconvergent at the mesh points and element midpoints, while the derivatives (which you use to compute stresses) are superconvergent at the 2 Gauss integration points. The derivation of the superconvergent results for 2D and 3D is more problematic, and restricted to meshes with high degree of symmetry. Nevertheless, this is an excellent theoretical treatment of the subject.

The reason I think I 'missed' this result, that 'stresses are more accurate at the integration points,' is that my background is in p-version, which uses higher order shape functions, and generally does NOT use the isoparametric elements. It seems that this result "stresses are more accurate..." is confined mainly (but not exclusively) to isoparametric elements of the type most FE analysts use (which doesn't make the isoparametric elements inferior of course, it's just a property specific to those element types). I still think it is incorrect to state, though, that 'stresses are superconvergent at the Gauss points' in general for all element types; still it might be true, perhaps it just hasn't been proved for all element types as far as I can tell.

 

[prost]

Incidentally, I doubt very much my use of 1D elements to illustrate that the stresses aren't actually computed directly until after the solution (the displacements) are found was 'specious'--the derivation of 1D, 2D and 3D finite element matrices have many many similarities that illustrate the same points: one starts with stress (or force) equilibrium, multiplies by a virtual displacement field, does some fancy integrations, assumes the functional form of the displacements, and voila! one derives the finite element matrices. The principles are much the same, the derivations much the same, the results much the same--you are calculating displacements, and compute stresses either directly with the derivatives of the displacement field or with a projection technique, but only after the displacement field is known. ALL authors of finite element texts I've seen start with 1D to illustrate the fundamental concepts, and move on to 2D and 3D once those concepts are understood for 1D elements. Of course not ALL fundamental concepts can be illustrated in 1D, but most can.

 

[rb1957]

this has been a very interesting technical discussion (almost a good, ol' fashioned "bun-fight"), but practically, how different are the stresses at the gauss points compared with the stresses at the nodes ?

particularly if you consider that "most" of the time we average the stress at the node (using the different nodal stresses from the different elements) ? ... most of the time i use element centroidal stress.

 

[btrueblood]

Prost,

The 1D arguement is specious (meaning fallacious), because it can be argued either way. I.e. I can just as readily argue that you must know the stress before you can start computing the displacements, it's just a multiplication factors, so who is right - depends on which way you started. You choose to start from displacements, I could just as easily derive the element model starting from stresses.

But whatever. You found my reference, read it, and it said what I said it said.

BTW, you're welcome.

 

[rb1957]

there were a few stress-based elements in the early days of FEM (ie assumed a stress distribution and worked from there) but (a bit like beta and VHS) the "other" displacement-based system won out.

 

[feajob]

Prost,

Following is a journal paper which discusses about the accuracy of stresses in FEA. You will see much more references in this paper.

The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique
O. C. Zienkiewicz, J. Z. Zhu

International Journal for Numerical Methods in Engineering, Volume 33 Issue 7, Pages 1331 - 1364


Following link also provides some more reference papers:

http://www.geocities.com/fea_tek/ali/index_pub.html

A.A.Y.

 

[prost]

Thanks feajob. This reference I missed. Was this the source of the "superconvergent patch recovery" technique I've heard about? I've seen a few papers recently that discussed proofs for particular cases.

 

[prost]

I still don't think you need to know anything about the stresses to compute the displacements. You are enforcing equilibrium of stresses, but only in a global sense through the integrals of the principle of virtual work, which is one reason why they call the principal of virtual work "the weak form" compared to the stress equations of equilibrium ("the strong form") as stress equilibrium is enforced point wise. You start with stress equilibrium, do some integrations and other fancy math, and what results looks pretty far away from your original stress equilibrium equations to me.

Is there a guarantee that because the stresses are superconvergent somewhere (for the sake of example, say they are superconvergent at the Gauss points) that equilibrium of stresses is obtained at those Gauss points?