It's common to divide errors into two broad areas, which I will refer to here as Modeling errors and Numerical errors( Does anyone has better descriptive terminology ? ). The two areas aren't really exclusive, but it's convenient to think of them as such.
Modeling errors refer to how you choose to simplify/represent the real world problem without regard to the computer/numerical issues. For example, taking a mult axial, time varying load and representing it with a single peak force. Or assuming that a press fit assembly is represented with "welded" or "ridgid" contact without accounting for the press-fit stresses. Proper selection of material properties is also included in this area. I will also add that you should have a good plan of how you will use the results before you get them. Will you be comparing to a previous design or to an engineering failure criteria such as yield stress or von Mises (I think that not having a plan for the results is one of the most common mistakes made in using FEA)? "Modeling" errors are where an experienced and knowledgeable client may be better suited than a consultant analyst at judging the fitness or acceptability of the assumptions. Your design team and analayst should discuss the modeling representation used and their appropriatness. Ideally, the analyst could show that assumptions (such as ignoring small off axis loadings) make very small contributions to the result.
Numberical errors refer to problems with descritizing the continuous model for computer analysis and numerical issues that may arise (especially in nonlinear analysis). For example, the number and type of elements used to represent the real world device, or the type of solver used in a nonlinear analysis as well as the time interval in transient analysis. These errors are more difficult for those not familiar with FE to check. As mentioned previously, a mesh density study will help show sufficient resolution. One should check for the continutity or smoothness of the depenedant variable (eg are the stresses/displacements "smooth"). Beware of results plots that are averaged because averaging can hide meshing problems as well as make judgements of stress continuity more difficult. Exagerated plots of the deformations are a good tool for this. Also, statistics of the element quality can be helpful, though poor element shape does not automatically mean the result is erroneous. As mentioned in modeling error section, in an ideal situation, your analyst could show the effect of descritization or averaging by demonstrating the effects of each parameter to the overall result (eg., that the difference in stress between 1000 and 10000 elements is negligable).
Unfortunately, its not so easy to look backwards at a result and know if errors are modeling or numerical related. Sometimes, the two errors can acutally cancel each other. Doing some manual calculations (combined with experience) is one of the best ways to validate a model and you should always include this. Trying picking some "easy" locations (the geometry is simple, or the loading components are simple) and see how close the FEA matches what you get. You can also try bounding the calculations to get a reasonable range that you expect the stresses should fall within. Real world testing is the gold stanadard, but that can have a whole other set of errors that sometimes make it less desireable than a hand check.
A final word of wisdom: I read from another FEA user (I think on Eng-Tips, I apologize to the individual for not providing credit) that his goal was to provide such detailed error checking that for someone to prove him wrong would be too expensive/timely to even try. This maybe the best advice I've seen for FEA.
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Implantable FEA for medical device manufacturers