As stated, the main challanges with modelling fracture mechanics problems, via Finite-Element [FE] and Boundary-Element [BE] analysis occur from the presence of an infinite-stress at the crack-tip. Sufficient accuracy can typically not be achieved, even by utilizing a very fine-mesh or applying higher-order elements. Due to the fact that the nature of the singularity of the crack-tip is known, unique crack-tip elements can and should be modelled that have built-in singularity. These elements have previously been utilized and modelled by others, via Finite Element Analysis and other methods to determine the Stress Intensity Factor. Other researchers, in the recent past, have clearly shown that the crack-tip elements are actually unnecessary, due to the fact that the discussed singularity can be easily incorporated by moving critical nodes on the isoparametric quadratic elements, so that the midpoint of the subject element is typically moved to the quarter point closer to the crack-tip, in order to create the desired stress singularity. Through the useage of these elements, the amount of mesh-refinement that is needed and necessary at the crack-tip, in order to obtain accurate values of the stress intensity factor, is significantly reduced.
How much mesh-refinement for three dimensional (3-D) structural-components is typically suggested at or near the crack-tip to generate verry accurate results. Are you utilizing any other specific methods to refine the mesh even further that work for you?
Christian C. Steputat
If you want to use refinement to get an accurate solution, you need an error estimator to guide the refinement procedure. How many times to refine is depends on the percentage of errors that the user wants to achieve. For more details you can read the book of Zienkiewicz “The Finite Element Method: Its Basis and Fundamentals”. Have a look in chapter 13 “Errors, recovery processes and error estimates” and chapter 14 “Adaptive finite element refinement”
This question is not unique to your problem. That is, other problems also present similar challenges with grid refinement. As you refine a grid, you may or may not see a minimum or maximum or clear asymptotic approach to a single value. Examples are presented in the literature where this is the case, for instance, 1000, 2000, 4000, 8000 elements yielding results that follow a simple pattern. Researchers often stop refining the grid when they run out of computer resources and declare that it is good enough! This is particularly difficult when you're studying something like a crack or other discontinuity. You cannot model an infinite stress and no real material can sustain one. Ultimately you violate the assumption of a continuum and get down to the grain or molecular level.
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