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Comparison of the Projection Methods
The projection methods FIE, GSB and MLS were applied to similar configurations and at least a tentative comparison of their relative merits is possible. The projection mechanism of the FIE method is completely in line with beam theory in regions
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Table 1 Memory consumption and execution time of the ACM with the available projection methods. Runs were carried out on a single 3.0 GHz Intel Xeon processor. The memory requirements include the storage of the system matrices and the solver workspace.
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where the beam axis is straight and does not have intersections. For the investigated test configuration, this region is the outboard part of the wing. In the vicinity of the beam kink the choice of the projection parameters Діть and dljmit has a moderate influence on the local shape of the deformed wetted surface. Correct or optimal values for either are hard to ascertain, and the default values are the results of experience. Approximate values for the width of the intersection region aljmit can be determined by common sense, but the choice has a more profound effect on the shape of the wetted surface. While not shown here, fundamentally the same behaviour regarding alimit can be expected for the MLS method. This method has the advantage that the number of support points n$ and the radius of the additional support points rFB can be varied over a large range with only marginal effects on the shape of the deformed wetted surface, independently of the polynomial order of the interpolation function. Compared to the FIE results the deformation distributions are slightly smeared during projection. Further differences to the FIE results are visible in the vicinity of kinks where the FIE method does not present a valid absolute reference either. Finally, the GSB method exhibits the strongest dependency on the choice of projection parameters. A high polynomial order of the global contribution to the interpolation function is beneficial, as it should already represent the deformation field as well as possible. The local RBF contributions have to make up for the difference between the supplied deformation distribution and its global approximation, so that the placement and number of supports bear a special importance.
From the user perspective, any projection method should not only be robust and deliver accurate results, but also have low computational resource requirements. The FIE method needs the least memory because in the implementation used in the ACM it does not store the projection matrix explicitly. The MLS method does so, but the resulting matrix is sparse. Discounting additional entries due to interpolation in intersection regions, the number of non-zero entries is nCFD x N$ = nCFD x NM x (nFB +1). With the GSB method though, the projection matrix is dense and the number of non-zero entries is the product of nCFD and the total number of support points. It is also the method associated with the greatest numerical effort. In Table 1, typical run-times and peak total memory requirements are summarised for the test cases treated in Chapters 4.1.3 and 4.2. With the shell model, the memory requirements are dominated by the structural system and preconditioning matrices. The overhead for the explicit storage of the projection matrix of the MLS method is not significant, as opposed to the beam model test case.
