Machine Design

# Test Your Foundation in Finite Elements

Paul Kurowski
President
ACOM Consulting

Although finite-element analysis has gotten a lot easier to handle over the past several years, users still must be aware of the science behind the software. Plenty can go wrong and the notion of accurate pushbutton analysis is still far off.

Over the years I have attended many finite-element- analysis courses and conferences, and listened to people involved in FEA one way or another. This small sample of statements from them reflects many common misconceptions. In many cases the statements are formulated imprecisely, as they were posted to me. They are randomly listed and some topics are discussed more than once.

The scope of FEArelated problems here is limited to structural and thermal analysis. The list, however, does not cover all FEA problems, and in no way can it replace a textbook or an FEA course. To test your knowledge of the FE science, try this simple quiz.

ANSWERS TO THE FEA TRUE-FALSE QUIZ
1. Usually true, if “better results” means more “accurate results” in terms of convergence error. However, this does not imply that we should always build fine meshes. Finer meshes are also more expensive, so a happy medium would combine acceptable accuracy with acceptable cost.

2. True, when “better results” means lower-convergence error. The finer the mesh, the lower the convergence error. Convergence error is calculated by refining the mesh and comparing results from two consecutive refinements. Almost any local or global measure can be used as convergence criterion. These include displacement of a particular node, maximal Von Mises stress, global strain energy, energy norm, or what have you.

3. False. Only structurally significant (or thermally significant in the case of thermal analysis) details should be modeled in the FEA model. Small and structurally insignificant details such as chamfers and draft angles unnecessarily complicate models and waste analyst’s time. Geometry coming from CAD usually includes a lot of details that should be deleted from the finite-element model. Understanding how the part will behave under load helps distinguish between significant and insignificant details. Some details such as sharp corners result in high stress concentrations. In short, CAD works with realistic geometry, FEA works with FEA specific geometry. Only on rare occasions are those two geometries identical.

4. False. Solids give a representation of model geometry only in a visual sense by making impressive images. Solid elements do not guarantee the best or even good results. Quite often, particular geometry does not lend itself to modeling with solid finite elements. An example would be a casting with thin walls. Properly modeling the wall with solid elements requires several layers of elements across the wall thickness. Doing so would result in huge models that are expensive or impossible to solve. Before using expensive solid elements, always consider more efficient shell elements.

CAD to FEA interfaces provide solidmodel geometry which can only be meshed with solid elements, unless it is reworked, such as creating midplane surfaces and meshing them with shells. It is unfortunate that CAD-FEA interfaces encourage the use of solids in cases where solids should not be used. Automeshers working on thin or solid geometry may place only one element across a wall thickness producing a terrible model when using h-elements. A one-elementthick wall will not model bending and will report greatly underestimated stress. Errors on the order of 500% are possible. This situation is less dramatic if we use p-elements since one p-element is usually sufficient to model bending.

5. False, of course. More expensive software may be easier to use and may produce more impressive plots. Good (correct results) depends on the user’s skills. A top-ofthe- line package may produce terribly wrong results, and vice versa.

6. True and false at the same time. Automeshing is certainly a time saver because it works much faster than mapped or truly manual meshing. However, automeshing most likely produces tetrahedral elements which are less efficient. Presently, few automeshers build brick elements. See answer 7 for more details.

7. False, for goodness sake. Automeshers know nothing about stress analysis. All they can do is fill up a geometric volume with elements. It is the user’s responsibility to assure that the mesh is refined where stress concentrations are expected, that there are enough elements across members in bending, and so on. For more difficult geometry, automeshers tend to produce terribly distorted elements and place them with no regard for laws of mechanics, such as one tetrahedral element across the wall in bending. Automeshers tend to work more reliably with p-elements, which can take more distortion and accurately model bending with only one element across.

8. False. First of all, who said that FEA is highly accurate? It certainly can be when properly used. Most FEA programs use double precision arithmetic for lower numerical error. In most cases, so-called computer accuracy or more precisely the round-off error

is small when compared with other errors like convergence error and modeling error. 9. False. No error means only that the model is correct from the solver’s point of view. Solvers happily run the most horrendous models as long as they do not run into numerical problems.

10. False. A single run provides results with unknown error. The error may be low but it is unknown. Without previous experience with similar models, analysts must run several mesh refinements to calculate convergence error and estimate solution error.

11. False. Even when using a fine mesh, analysts still do not know the error. In fact, building a fine mesh that provides an estimated solution error of, e.g., 0.1% (although we do not know that it is 0.1%) wastes time and money if all one really needs is a 10% error.

12. True in terms of convergence error. Higher-order elements in place of lower order elements are practically equivalent to mesh refinement. When comparing two otherwise identical models — one with first-order elements and the other one with secondorder elements — the later will provide more accurate results in terms of convergence error. Second-order-element models also converge faster so fewer mesh refinements are necessary for error analysis.

13. Generally true. Many tetrahedral elements or tets do not benefit from certain enhancements in element definition that make hexahedral (brick) elements behave like higher-order elements. Tets usually require more elements to properly model bending.

However, the foregoing does not mean tet elements produce wrong results when using enough of them. Using any kind of element requires understanding the basics of its formulation. Analysts need to know what the element can and cannot do, such as whether or not it models constant or linear (or quadratic) stress distribution within its volume, supports nodal rotations, or which elements can be combined in one mesh.

14. True in most cases. However, a first “rough” mesh still needs sufficient density to detect stress concentrations. When the element size is larger than the size of a hot spot, the stress concentration will not show up and analysts will know they should have refined the mesh in that particular location.

15. False. Strain gages may be placed in a spot that is modeled correctly in the finite-element model. However, correlation in one or more locations does not guarantee that everything is fine with the model. However, the opposite statement is true: when FEA does not correlate with experiment, then something is definitely wrong, either with the model, the experiment, or both.

16. False. The best mesh is the most coarse one that still provides acceptable accuracy. The “finest possible” mesh is simply a waste of time when there is no use for higher accuracy. Also, even the finest possible mesh gives no indication of error.

17. Generally true, if you use a reputable software and your analysis involves typical cases. Still, benchmark tests are recommended to get the feel of the program, particularly when applying it to unusual models or to unfamiliar conditions.

18. False. Degenerated elements tend to be too stiff and they affect the global model stiffness. In other words, they “pollute” the model. The pollution may propagate to the area of interest and render erroneous results.

19. True. In most cases nodal displacements converge faster than, for example, stresses. Still, using a coarse mesh for deflection analysis should be justified by the results of convergence analysis.

20. True. Modal analysis provides natural frequencies and modes of vibration which are of a global nature as opposed to local measures such as stress concentrations.

21. True. Results come in 10-digit numbers. Every conceivable bit of information on displacements and stresses can be obtained along with impressive animated plots and graphs. It’s too easy to forget that all these results often rest on crude assumptions like material properties, loads, supports, modeling simplifications, and finite-element discretization error.

22. True. Geometry is the most intuitive input and can be assessed by eye. Loads are relatively easy to relate because they are expressed in numbers. Boundary conditions, or supports in the case of structural analysis, are the most vague, and perhaps because of that, users often neglect the proper definition of boundary conditions. Most people simply model them either as perfectly rigid or as hinged supports, while the truth may be somewhere in between.

24. True. Loads, for instance, may not be the same. Finite-element models may also ignore their own weight, residual stresses, and surface-finish factors.

25. True. In the hands of an unskilled but enthusiastic user the FEA can be an expensive toy. Depending on the importance of the analysis it may also be an outright dangerous tool.

26. True. Some people say they use FEA for everything for better results. These poor folks are wasting their money. Other methods such as longhand calculations, engineering tables, math packages such as TK Solver, or testing are often less expensive, faster, and more reliable.

27. False. H-element software produces data of interest. But after a single run there is no convergence error analysis. P-element software produces data of interest in an iterative process and provides results complete with convergence error analysis. A single run of h-element software may take shorter than the iterative solution performed by p-element software. However, the proper basis for comparison of speed between h and pversion software is to compute data of interest and verify that the associated error is small. This basis for comparison shows that p-elements are actually faster. P-element software automatically generates a convergence error analysis. H-elements require that users perform several mesh refinements which may be time consuming.

 ANSWER TRUE OR FALSE TO THE FOLLOWING: 1. The finer the FE mesh on a model, the better the results. 2. A finer mesh gives better accuracy. 3. Geometry should be represented with as much detail as possible. 4. Solids give the best results because they accurately model the geometry. 5. Better (more-expensive) FEA software gives better results than lessexpensive packages. 6. Automeshing is better than manual meshing. 7. An automatic-mesh generator reduces meshing to a pushbutton operation. 8. The high accuracy of FEA results comes from the high processing accuracy of the computer. 9. When your FEA software reports no error, the solution will be correct. 10. You do not really need an error estimation. FEA is always accurate enough. 11. You should always make a fine mesh so you do not have to worry about error. 12. Higher-order elements give more accurate results. 13. Tetrahedral solids are too stiff and should be avoided. 14. Make a coarse mesh first to find stress concentrations, then refine it as needed. 15. When FEA results correlate well with strain-gage readings, for example, FEA results are OK. 16. Always make the finest model possible. Disk space should be your limit. 17. All major FEA codes have been extensively tested so users need not run benchmark models. 18. Analysts may ignore degenerated elements as long as they are far away from stress concentrations. 19. If you want to know only deflections and do not care about stresses, then you can make a coarse model. 20. Modal analysis can use a coarser mesh than stress analysis. 21. Finite-element models offer a deceiving level of detail. 22. Model geometry is the most readily controlled of all data, loads less so, while boundary conditions are the most difficult to control. 23. The most severe modeling errors are most often made assigning boundary conditions. 24. Test data always contain errors and may be inconsistent with FEA assumptions. 25. Incompetent analysis gives, at best, unreliable results. At worst, it is positively misleading. Bad FEA lets users misplace trust in the design. 26. FEA is often excessive technology. It’s similar to using a hammer where only a fly swatter is needed. 27. P-elements are slower than h-elements.