In this section, we verify that the conclusions made in the one-dimensional analysis extend to the two-dimensional case. The test case is solved using quadrilateral and triangular elements. The quadrilateral element mesh is structured, which provides an illustration of the aliasing error. For the analyses which follow, both downstream flow and upstream flow cases are considered. The downstream data demonstrates the aliasing affect, but is provided mostly for the sake of interest. The upstream case must generally be accounted for when designing finite element models of aeroacoustic problems, due to the shorter wavelengths that are present. Selection of the preferred method for further development will depend on the upstream analysis. Examples of the meshes used are given in Figure 4.8.
The L2 error, condition number, and number of non-zero entries are used as mea-
sures of performance. For the data that follows only the cubic order (P = 3) has been analysed, the reason being that only this order of the Hermite shape functions is avail- able in 2D. This section will be concluded with the identification of the method chosen for further development.
0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 x y 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 x y
Figure 4.8: Examples of meshes used for the two dimensional analysis: h = 0.1. Left:
Structured quadrilateral mesh. Right: unstructured triangular mesh.
4.3.1 Downstream Flow Case
To generate the data presented here the quadrilateral and triangular meshes given in Figure 4.8 are used. To demonstrate the aliasing effect, and the behaviour of the methods when subjected to a downstream flow (M = 0.6), the frequency of interest is gradually varied from ω = 7 to 40. The results are presented in Figure 4.9.
4.3 Two-Dimensional Analysis 101 10−3 10−2 10−1 100 101 Frequency, ω L 2 error (%) Hermite P−PUM, Bernstein Lobatto 101 103 104 105 106 Frequency, ω Condition number
Hermite, Bernstein, Lobatto
101 10−4 10−3 10−2 10−1 100 101 102 103 Frequency, ω L 2 error (%) P−PUM Bernstein, Lobatto Hermite 101 104 105 106 107 108 109 1010 1011 Frequency, ω Condition number Hermite P−PUM Bernstein, Lobatto
Figure 4.9: Convergence and conditioning properties of the methods, for the downstream
flow case: M = 0.6, h = 0.1, c0 = 1, ρ0 = 1. Top: quadrilateral elements. Bottom:
triangular elements.
4.3.1.1 Quadrilateral Elements
It can be seen that the convergence plots exhibit peaks. The peaks correspond to internal numerical resonances within the elements. This is the aliasing error which was introduced in Section 4.1.2.
In the asymptotic region the L2 error of the methods converges like O ωP +1. The
Hermite method returns the highest levels of error, the polynomial partition of unity and Bernstein methods give identical error levels, and the Lobatto method provides the lowest error levels.
4. ASSESSMENT OF HIGH ORDER METHODS
expected from the 1D observations (see Section 4.2.3). For clarity, the condition number of the polynomial partition of unity method is not included here as it has a level of 1020. We note that, when using a structured mesh the linear dependency issue
reported in Section 3.3.3 is not remedied by the removal of the enrichment functions at the boundary. The condition numbers of the remaining methods are quite similar, although asymptotically the Bernstein method produces the best conditioned systems.
4.3.1.2 Triangular Elements
If we now consider the convergence plots obtained using the triangular mesh we see that the error peaks are not as pronounced as they were in the quadrilateral data. This is due to the non-uniformity of the unstructured triangular mesh, which tends to broaden and flatten the peaks. However, the effect of the aliasing error is still observable.
The L2 error of the partition of unity method behaves like a linear polynomial
system. The Bernstein and Lobatto systems produce identical error levels, but the Hermite system incurs the lowest level of error.
The condition number of the Hermite matrix is the highest, closely followed by that of the polynomial partition of unity matrix. The condition numbers of the Bernstein and Lobatto matrices are similar, but in the asymptotic region the Lobatto matrix is the best conditioned.
4.3.2 Upstream Flow Case
The convergence and conditioning plots for the upstream case (where M = −0.6) are given in Figure 4.10. To obtain these plots the frequency is kept constant (ω = 20) and the number of quadrilateral and triangular elements is gradually increased. Dashed lines are included on the plots to indicate the orders of convergence of the various methods investigated. Note that in the upstream case there is no aliasing error [28], as the waves are well resolved in all directions (which is not true for the downstream case).
4.3 Two-Dimensional Analysis 104 105 106 10−1 100 101 102
Number of non zeroes
L 2 error (%) P−PUM Bernstein, Lobatto Hermite 104 105 106 105 1010 1015 1020
Number of non zeroes
Condition number
P−PUM
Bernstein, Hermite, Lobatto
104 105 106
10−1 100 101 102
Number of non zeroes
L 2 error (%) P−PUM Bernstein, Lobatto Hermite 104 105 106 104 106 108 1010 1012
Number of non zeroes
Condition number
Hermite
P−PUM
Bernstein
Lobatto
Figure 4.10: Convergence and conditioning properties of the third order functions, for
the upstream flow case: M =−0.6, ω = 20, c0= 1, ρ0 = 1. Top: quadrilateral elements.
Bottom: triangular elements.
4.3.2.1 Quadrilateral Elements
From the error convergence plots we see that the partition of unity system is the most expensive to store. The Bernstein and Lobatto systems return identical error levels as a function of the number of non-zero entries. For the range of numbers of elements considered, the Hermite system is the most efficient, at least in terms of matrix storage requirements.
We see that the Bernstein and Lobatto solutions converge like O Nnz−3, where Nnz
is the number of non zero entries. The partition of unity error converges like O Nnz−2.8, while the Hermite error converges like O Nnz−2.4. As expected, the partition of unity matrix is severely ill-conditioned, whereas, in the asymptotic region, the Lobatto matrix is the best conditioned.
4. ASSESSMENT OF HIGH ORDER METHODS
4.3.2.2 Triangular Elements
The error of the partition of unity system converges like O N−1.5
nz . The Bernstein
and Lobatto error convergence plots are identical and converge like O Nnz−3. For the range of non-zero entries considered, the Hermite system is the most efficient, in terms of storage requirements. Asymptotically, it converges like O Nnz−2.4.
The Hermite matrix has the highest condition number, followed by the partition of unity matrix. In the asymptotic region the Lobatto matrix has the lowest condition number.
4.3.3 Summary of Two-Dimensional Analysis
Based on the upstream flow case analysis, it can be concluded that for the quadrilat- eral elements the polynomial partition of unity method produces systems which are inefficient and ill-conditioned. For triangular elements the system is better conditioned (perhaps due to the unstructured nature of the mesh, which reduces the effect of the linear dependency issue) but it does not converge as expected. These shape functions will not be considered for further development.
The Hermite elements appear to be the most efficient in terms of matrix storage, but they have a high condition number when triangular elements are used. Furthermore, this analysis has considered only one polynomial order (P = 3), due to the limited availability of higher order triangular Hermite functions, and without linear or quadratic functions any adaptive scheme based on these functions may be needlessly expensive when accurate geometry description is required. These functions, despite their elegance, will not be considered any further.
In the upstream case, the Bernstein and Lobatto systems produce identical error levels. However, the Lobatto elements are better conditioned. (Note that this differs from the findings of Petersen [121], however, they use an iterative solver and do not include flow.) The Lobatto functions are hierarchic, which enables efficient system assembly when using an adaptive order process to set-up a model, and there are readily available functions for all commonly used element types. The Lobatto functions have been chosen for further development.
4.4 Analysis of Lobatto Shape Functions