Premisas de diseño

The company sponsoring this project, LMS International, have implemented the pro- posed adaptive order scheme into their aeroacoustic solver, SysNoise. This has enabled tests of the performance of the adaptive scheme in three dimensions. The results of these tests are presented in this section. Four meshes with increasing non-uniformity have been generated, and are used to test the adaptive scheme’s ability to handle dis- torted meshes. The meshes are presented in Figure 5.19. Three predefined error levels were chosen to provide users of the solver with coarse (15%), standard (5%), and fine (0.5%) resolution options. Look-up tables for these levels were created using the data presented in Figure 5.14.

5. DEVELOPMENT OF AN ADAPTIVE ORDER SCHEME

Figure 5.19: Meshes used to test the adaptivity scheme in 3D. Top left: h = 0.2, top right:

h1/h2= 0.2/0.04, bottom left: h1/h2= 0.2/0.02, and bottom right: h1/h2= 0.2/0.05

5.3.2.1 No Flow Case

The unconvected wave equation has been solved. The L2 error as a function of the frequency is given in Figure 5.20. A general observation is that the actual error, for all meshes and for each of the accuracies (coarse, standard, and fine), is bounded by the desired error level. It can be seen that the error plots exhibit very little variation with frequency, until the mesh frequency limit is exceeded.

5.3 Illustration of Performance of Proposed Estimator

We observe that this limit depends on the accuracy chosen, which is expected as a higher accuracy will require elements with higher orders, and thus the highest available order will be required on a majority of the elements at high frequencies. The flat error plots also indicate that a minimal amount of unnecessary computational effort has been incurred. As was observed for the 2D results, the error levels produced by the uniform mesh exhibit dips; these are caused by the uniformly increasing polynomial order of the elements with increasing frequency. The results obtained from the mesh with an element size ratio of 4:1 exhibit ‘bumps’ in the error. This mesh has a much higher number of small elements than the other meshes (as the entire boundary is composed of small elements), and thus the changing of the polynomial order of these smaller elements causes the bumps in the error.

In Figure 5.21 the maximum, most common, and minimum polynomial orders used to solve the system, to a ‘standard’ accuracy, are given as a function of frequency. For the uniform mesh (ratio 1:1) we note that the order is not always the same on every element. Since this is an unstructured mesh the elements are not all exactly the same size. In the case of the mesh with an element size ratio of 4:1, we see that the maximum order is much higher than the most common and minimum orders. This is due to the high number of the smaller elements in the mesh, which are able to give good accuracies using lower orders. Note that the incremental changes of the minimum and most common orders coincide with the bumps in the error (referred to in the previous paragraph).

In Figure 5.22 the memory required for matrix factorisation as a function of fre- quency is given. As expected, the uniform mesh requires the least amount of memory, and this is because the uniform mesh has the highest number of large elements. This in turn means it has the lowest number of elements, but also that the order on those elements is higher. As already shown, using higher orders is optimal due to the re- duced factorisation memory requirements. For the highest frequencies the mesh limit is reached, as indicated by the plateau in the memory needed. The distorted mesh requires the most memory, as it has the highest number of low-order elements. This comparison confirms the efficiency of using higher order elements. In Figure 5.23 the time required to solve the system is given as a function of frequency. The same con- clusion can be drawn as for the memory requirements - it is most efficient to use large elements with higher order interpolation functions.

5. DEVELOPMENT OF AN ADAPTIVE ORDER SCHEME 10 20 30 40 50 60 70 80 10−2 10−1 100 101 102 Frequency, ω L 2 error (%) 10 20 30 40 50 60 70 80 10−2 10−1 100 101 102 Frequency, ω L 2 error (%) 10 20 30 40 50 60 70 80 10−2 10−1 100 101 102 Frequency, ω L 2 error (%) 10 20 30 40 50 60 70 80 10−2 10−1 100 101 102 Frequency, ω L 2 error (%)

Figure 5.20: L2 _{error as a function of frequency. Coarse (- - -). Standard (—–). Fine}

(...). Mesh ratios: top left: 1:1, top right: 5:1, bottom left: 10:1, and bottom right: 4:1.

10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 9 10 Frequency, ω Polynomial order 10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 9 10 Frequency, ω Polynomial order

Figure 5.21: Order as a function of frequency. Maximum order (). Most common

5.3 Illustration of Performance of Proposed Estimator 100 101 10−2 10−1 100 101 Frequency, ω

Factorisation memory (Gbytes)

100 101 10−2 10−1 100 101 Frequency, ω

Factorisation memory (Gbytes)

100 101

10−1 100 101

Frequency, ω

Factorisation memory (Gbytes)

100 101

10−1 100 101

Frequency, ω

Factorisation memory (Gbytes)

Figure 5.22: Memory as a function of frequency. Coarse (- - -). Standard (—–). Fine

(...). Mesh ratios: top left: 1:1, top right: 5:1, bottom left: 10:1, and bottom right: 4:1.

10 20 30 40 50 60 70 80 101 102 103 Frequency, ω Factorisation time (s) 10 20 30 40 50 60 70 80 101 102 103 Frequency, ω Factorisation time (s)

Figure 5.23: Time as a function of frequency. Coarse (- - -). Standard (—–). Fine (...).

5. DEVELOPMENT OF AN ADAPTIVE ORDER SCHEME

5.3.2.2 Flow Case

In this section the flow case is solved using the adaptive scheme with the standard accuracy (5%), for flows with Mach numbers 0.4, 0.2, −0.2, −0.4, and −0.6. The resulting L2 errors are given as functions of the wavenumber in Figure 5.24.

A general observation is that the upstream error levels are very similar in level, although there is a noticeable increase in error with increasing Mach number. This slight increase may be caused by either pollution error or the boundary conditions of the problem - neither of these possible errors are accounted for by the adaptive order scheme. Furthermore, the scheme does not account for the wave direction across an element, which means that in the presence of flow the worst case scenario must be accounted for, which is the upstream flow case. A consequence of this is that in the presence of downstream flow the method becomes inefficient. This can be seen in the error plots shown. Waves in the downstream case are over-resolved, and thus incur unnecessary computational expense.

As in the no flow case, the uniform mesh error exhibits dips which are associated with the uniformly changing orders of the elements. In the downstream case this mesh is more expensive to solve than the other meshes due to the increased order of the element. The remaining meshes exhibit similar results; the upstream cases exhibit little variation in error with frequency, while the downstream cases are computationally expensive. The only way to make the adaptive scheme more efficient on elements that are subject to both upstream and downstream flow is to have advanced knowledge of the wave direction across each element. However, this is no trivial task, and is left for future work.

The factorisation times as a function of the wavenumber for the uniform mesh and the mesh with an element size ratio of 4:1 are presented in Figure 5.25. These results support the observations already made: the upstream cases take similar amounts of time to be solved, while the downstream cases are inefficient, and incur unwanted additional computational expense.