To calculate the optimium coefficients that produces crack growth that matches existing crack growth data we need to supply the following information:
- a crack growth model corresponding to each crack measurement file that including beta, stress, start and end cracks size, dadn model
- loading sequences used for the crack measurements
Each line in the optimisation file is an easigrow command line that will generate a crack growth curve along with a measurement file of crack growth data that it tries to match.
Here is a sample listing of crack growth measurements obtained from a microscope.
Item X (mm) Y (mm) Z (mm) Radius (mm)
1.00 0.0000 0.0000 -0.0000 0.0000
2.00 0.1280 0.1638 0.1991 0.2079
3.00 -0.0128 6.0006 0.0009 6.0007
4.00 -3.7212 6.2297 -0.0620 7.2565
5.00 -3.7212 6.2297 -0.0620 7.2565
6.00 -3.7572 6.2297 -0.0666 7.2750
7.00 -3.7910 6.2280 -0.0670 7.2911
8.00 -3.6810 5.7518 -0.1105 6.8288
9.00 -3.6810 5.7518 -0.1105 6.8288
10.00 -3.7184 5.7518 -0.0871 6.8490
Note, these measurements are taken at the same point within a block or feature on the fracture surface. In this case, because we are only dealing with blocks, the exact load line the measurements were made at is not important. Unless otherwise specified, the program will assume it is the first load line. Contiguous measurements are made until the blocks can no longer be identified. We indicate that a measurement is not contiguous with the previous block and is the start of a new run by pressing the measurement button twice. Here measurements 4 and 5, and 8 and 9 are both the same indicating the start of a new run.
This measurement file is turned into an input file that looks like
0.2079
6.0007
7.2565
7.2750
7.2911
6.8288
6.8490An example of the output from an optimisation run is shown below. The output shows how the optimisation is doing. Here we have used the Walker equation which contains three variables which are optimised, they are [0.00000001, 3, 0.5]. We also include the crack growth from 4 coupons, each with an associated crack growth file. The error term is calculated for each of the files and compared with the easigrow predictions using the initial set of parameters. In this case the initial parameters are the default values which are not very good.
There are a number of different optimisation methds but the simplest is Nelder optimsiation method which is used for this example. At the start of a Nelder optimisation the algorithm adds a small increment to each of the optimisation parameters in turn to determine which direction is best to begin the optimisation. Then it will head off in the most favourable direction adjusting the other parameters looking for directions that result in some reduction in the overall error. When the level of improvement in each step falls below the convergence threshold it will terminate the optimisation with a message saying it has converged.
The program may also terminate if it has exceeded the maximum number of iterations. The default value for the maximum number of iterations in an optimisation is 100. This can be overridden on the command line, as we have done here, increasing the maximum iteration limit to 1000. The initial error terms for each of the crack growth files are all in the region of 100--200 i.e. not a good prediction. Eventually, the solution converges and we have an optimised answer where the error between the optimised predictions and the measured crack growth are of the order of 20--30. A significant improvement. The final optimised parameters are [1.23759e-9, 2.0023, 1.8756].
Inside the program, the optimisation is performed on a non-dimensionalised version of the parameters, where the parameters are normalised based on their starting values. This makes the scaling of the parameters much closer to each other, which is better for most optimisation routines. The resultant normalised factors are [0.12376, 0.66743, 3.7511].
easigro --optimise optim --opt_max 1000 --opt_method Nelder -d walker:default