Requested Contours and GRef Influence

As mentioned in section 2, the J-Integral, that allows the stress intensity factor deduction, is calcu-lated by 2D closed contour integrals. In theory, the J-Integral is path independent, but numerically (or computationally) this is not true. Thus, different contours give different solutions for the stress intensity factor.

Abaqus ^R allows the user to introduce the number of contours to use for the stress intensity factor calculation, but what number should be introduced? Furthermore, it was observed that the package of requested contours is calculated for each of the generated points in the crack’s front.

The number of generated points, appears to be related to the number of mesh divisions in depth. For structured geometries, if the geometry is a parallelepiped and has i divisions in depth, there will be i + 1 point, which implies i + 1 packages of j contours, each one with a stress intensity factor estimation.

Against this backdrop, it was necessary to design a set of tests that could clarify various doubts. A large sample of analysis was needed in order to be able to understand how to tune the parameters to obtain good results.

Choosing then a SENT geometry, which was set with a height H = 90 mm, width W = 10 mm, crack length a = 2 mm, it was made the study of the computational burden associated to the mesh variations.

In terms of thickness, B = 1 mm, and only 2 divisions in depth were considered. For the elements, it was used hexahedral elements with first order accuracy, and reduced integration, since as it is shown later on this thesis, these characteristics are the best choice.

As previously stated, two sets of refinement were defined, Global Refinement and Depth Refinement, referred to from now on as GRef and DRef. In this set of analyses, the first will vary and the second will be fixed in 2 divisions. The partition scheme remains constant, so the increase of GRef , increases the mesh density around the crack.

Another question arises now. What will be the composition of the solution since each contour pro-vides a different value for the stress intensity factor? And which contours should be chosen? All of them or only some?

In all the existing Computed Assisted Engineering (CAE) solutions, regarding the stress intensity factor calculation, the number of contours for the stress intensity factor calculation is asked to the user.

In the classical method, 5 contours is always desirable [1]. As the XFEM is still a recent method, it was decided to also study the impact of the requested number of contours.

Hence, considering the aforementioned geometry, with GRef = 80, DRef = 2, and the traction tension unitary, different numbers of contours for the stress intensity factor calculation were requested.

The considered range of contours was [5, 10, 15, 20, 25] and for each contour, the error was calculated.

For the errors calculations, it has always been used a relative error expression as the written in equation 4.1, where Kref is the reference stress intensity factor obtained from literature.

relative=KI− Kref

Kref

(4.1)

The results can be appreciated on figure 4.5 where the first contour is always neglected.

Figure 4.5: Error evolution for different number of requested contours

It may be said that all the different plot lines oscillate in turn of the same final value. As can also be observed, it is confirmed that each contour gives a different value for the stress intensity factor. It is necessary to define how to calculate a final value. The fairest way, it is to make an average.

It was thought that this average could be a controlled one, by the exclusion of the values that affect to much negatively the solution, but it was decided to consider all the values for the average except the one for the first contour, which is always far from the solution. This decision is related to the fact, that if a random geometry is considered, for which the correlation for the stress intensity factor is unknown, it will be very hard to know what values can be excluded. In order to develop a strategy, it is important to work with the same rules that if a random geometry was the case under evaluation.

Having the average in mind, it was possible to derive table 4.1.

Table 4.1: Impact in the average error of the requested number of contours Number of Requested Contours [#] Average Error [%]

5 0.46

10 0.53

15 0.42

20 0.42

25 0.54

As the sample of contours growth, the average error becomes lower and a minimum is observed for 15and 20 requested contours. The variations in the error are very small (0.1%) but it was decided to ask 20contours in all further analyses, to observe the error evolution with the contour number.

More contours could be requested but it was verified that sometimes, for higher number of requested contours, the outer contours may give values for the stress intensity factor completely wrong. This makes sense because if a too high number of contours is requested, the outer ones, will have a radius relatively to the crack’s tip so big, that the solution is no more accurate.

For the data treatment, it is important to refer that each Abaqus ^R analysis, produces a very long output file, with all the kind of information, such as the elements and nodes number, the time taken by the analysis, among others, which makes the data treatment time consuming. Thus, it was programmed in C + + a code that takes as input the output file from Abaqus ^R, and processes all the information, isolating what is needed, and in the end, writes a new file. This feature proved to be a big jump in this thesis since it has saved hours of data analysis. The treated data were then entered into excel files, allowing the graphs and tables production.

After the study of what should be the number of requested contours, 6 different meshes were considered, consequential to 6 variations in the GRef value, which took one of the following values:

[10, 20, 40, 80, 160, 200]. For all these analyses, DRef = 2.

This set of values assigned to the global refinement conducted to the analyses characteristics pre-sented in table 4.2.

As can be seen in figure 4.6, the time evolution with the parameter GRef is exponential, indicating that the GRef parameter is time sensitive.

From table 4.2, it may be observed that the number of elements and nodes, increase very fast. From GRef = 160, the analyses already has many elements and are time consuming, becoming heavy since the analyses should be expedite for this particular case, which is a simple geometry. To more complex geometries, it is reasonable to expect rather longer runs, associated to heavier meshes.

For each value of GRef it is calculated 3 packs with 20 stress intensity factors. The 3 packages are consequential of the 2 divisions in depth considered for the DRef value. Each one of the 20 stress intensity factors is consequent of a J-Integral calculation, in one of the 20 considered contours.

Table 4.3, shows part of an excel file for a given refinement.

For each GRef value, it is extracted the time¹taken by the analysis, the number of nodes as well the

1Abaqus ^R gives different times for each analysis; the more important are the CPU time and the Wall-clock time. It is important to note that the chosen reference time for this thesis was the Wall-clock. It corresponds to the human perception of the passage of time from the start to the completion of a task. In this thesis, it was used a processor Intel i7, with 8 logic units and a CPU of 3.6 GHz, associated with a RAM memory of 8 GB

Figure 4.6: Time evolution with the GRef parameter

Table 4.3: Numerical study, files appearance

number of elements.

For each of the 3 generated points, the 20 stress intensity factors estimations are introduced and the respective error is calculated.

To each point, it is associated an average error. At last, the absolute average error is calculated, corresponding to the average of the three averages errors. In practise, this absolute average error, corresponds to the average error verified for a given GRef value.

It is important to note, that in the case of 2 divisions in depth, which produces 3 points, it was always the centre one who gave the best results. So far, the explanation has not been found, but it is the main reason which support the fact that all important validations in this thesis are done with DRef = 2.

Another interesting point is the evolution of the error, with the number of contour, for the different analyses (figure 4.7). The plotted values are extracted for each analysis from the second point, the more accurate values.

The more eye-catching point is the fact that the second contour produces the biggest error for all GRef values.

Then, the solutions for GRef = 10 and GRef = 20 divisions appear unstable. Their behaviour invalidates them completely. From GRef = 40, all solutions are cleaner.

The closest solutions are from GRef = 80 divisions. The times taken by the analyses of GRef = 160

Figure 4.7: SENT contour error evolution

and GRef = 200 divisions are already very high. The difference between the solution for GRef = 80 and GRef = 160 is 0.02% (table 4.4), however in computational cost (Wall-clock time), GRef = 160 is 31 times more expensive than GRef = 80 (table 4.2), which makes the analysis of GRef = 80 a reference for subsequent analysis.

Ultimately, the solution is converging, as can be appreciated on table 4.4. In short,the results are very good revealing a very good start for this XFEM evaluation and validation.

Table 4.4: SENT GRef influence results GRef [#] Absolute Average Error [%]

10 3.16

20 1.92

40 0.91

80 0.48

160 0.50

200 0.51