Chapter 3 Global Proper Orthogonal Decomposition and Re-
4.4 Numerical Results
the ROM shows a good agreement with that of the FOM. If the coordinates transformation in Eq. (4.15) is not executed the interpolation is not vaild.
The same is true for the position 2 shown in Fig. 4.3. On the other hand, if the sampling range increases, the slight difference occurs at 1,500∼2,000 Hz and 4,000∼5,000 Hz as shown in Figs. 4.4 and 4.5. This gap becomes larger as the sampling range increase to 24 mm (Figs. 4.6 and 4.7).
To examine the performance of the interpolated ROM, the errors occured by the sampling range are studied. In Fig. 4.8, the average relative error of eigenvalues from the first to eighth are presented according to the change of sampling range. The thickness of new operating point is 20 mm and the error increases nonlinearly. In addition, random thicknesses were set as new operating points. Total 1,000 samples were investigated and min, max and average errors are plotted in Fig. 4.9. Thus the interpolated ROM error can be specified by setting the sampling range.
Example 2. Cantilever plate
The second example in Fig. 4.10 is cantilever plate with 4 subdomains. The freqnecy response was examined in the range of [0:0.25:500] Hz, also the 201 snapshots of frequency response were taken in [0:2.5:500] Hz. Different from the previous membrane example, the stiffness matrix is a function of linear and cubic polynomial such that
K(µ) =µKshear+µ3Kbending. (4.43)
the sampling range and changed the polynomial order to interpolate. The thickness of new operating point is fixed to be 7 mm. Fig. 4.11 shows the results of linear interpolation. As shown in Eq. (4.43), the linear sampling yields poor results in overall frequencies. For the quadratic sampling shown in 4.12, althogh the frequency response becomes much better compared to linear case, still some misalignments are exists. The cubic sampling in Fig. 4.13 shows a good agreement to the FOM. For the Lagrange interpolation using cubic polynomial, most frequency responses except the range around 150 Hz shows poor results. As mentioned before, the interpolation of eigenvector is more strict than that of eigenvalues. However, the developed method using the moving least square accuratly predicts the frequency responses. Fig. 4.9 presents the comparison of the relative errors of eigenvalues. As expected, the increase of polynomial order results in the decrease of the errors of ROM.
However, for the 8th mode, the error does not decrease but slightly increases.
Because, the 8th mode is rotational mode in xy plane. The prediction of the rotational degree of freedom comes from the performance of the element used.
Therefore, in this plate bending problem, this problem is not significant. In fact, the error is less than 0.5 %, which is quite small compared to the other modes.
Example 3. Wing box model
The last example is wing box model with 8 subdomains showin in Fig. 4.15.
The geometry and material properties are the same to the one in Chap. 3.
The frequency range of the FOM is [0:0.05:100] Hz, and the 101 snapshots were taken in [0:1:100] Hz. Fig. 4.16 shows the frequency responses of the
interpolated ROM with the 4.5 mm overall thicknesses. The linear sampling was executed at the 4 mm for lower bound and 5 mm for upper bound, which results in 256 ROM constructions totally. Whereas the Lagrange interpola- tion was executed to interpolate the ROMs, the moving least square method was employed for the recovery process. The two responses show good agree- ments for all degrees of freedom. However, at 80∼100 Hz range, the some misalignment observed since the sampling is linear. If we use cubic inter- polation, more accurate results can be obtained, which also requires much more computations in off-line stage. This issues will be presented in Chap. 5.
In Fig. 4.17, the average of relative eigenvalue errors of 1,000 random sam- ples were computed within 5∼10 mm and corresponding probability density function (PDF) was obtained. Thus we can predict the error of interpolated ROM based on the PDF. If the sampling range increase, the relative error also increase as shown in example 1.
Table 4.1 Cases of sampling ranges Nominal: 20 (mm) Lower Upper Range
Case 1 16 24 8
Case 2 12 28 16
Case 3 8 32 24
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
−0.05 0 0.05
0.1 Sub. 4
µ4 Sub. 2
µ2
Sub. 3 µ3 Sub. 1
µ1 Impulse
(1) Frequency Response (2)
Figure 4.1 Cantilever beam with 4 subdomains of plane stress membrane element under tip impluse load.
0 1000 2000 3000 4000 5000
−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (dB)
(a) x
FOM PROM: Range 8mm PROM w/o coord. trans.
0 1000 2000 3000 4000 5000
−250
−200
−150
−100
−50 0
Frequency (Hz)
Magnitude (dB)
(b) y
Figure 4.2 Comparison of frequency responses of the FOM and ROMs at position (1): 8 (mm) sampling range.
0 1000 2000 3000 4000 5000
−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (dB)
(a) x
0 1000 2000 3000 4000 5000
−250
−200
−150
−100
−50 0
Frequency (Hz)
Magnitude (dB)
(b) y
FOM PROM: Range 8mm PROM w/o coord. trans.
Figure 4.3 Comparison of frequency responses of the FOM and ROMs at position (2): 8 (mm) sampling range.
0 1000 2000 3000 4000 5000
−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (dB)
(a) x
0 1000 2000 3000 4000 5000
−250
−200
−150
−100
−50 0
Frequency (Hz)
Magnitude (dB)
(b) y
FOM PROM: Range 16mm
Figure 4.4 Comparison of frequency responses of the FOM and the ROM at position (1): 16 (mm) sampling range.
0 1000 2000 3000 4000 5000
−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (dB)
(a) x
0 1000 2000 3000 4000 5000
−250
−200
−150
−100
−50 0
Frequency (Hz)
Magnitude (dB)
(b) y
FOM PROM: Range 16mm
Figure 4.5 Comparison of frequency responses of the FOM and the ROM at position (2): 16 (mm) sampling range.
0 1000 2000 3000 4000 5000
−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (dB)
(a) x
0 1000 2000 3000 4000 5000
−250
−200
−150
−100
−50 0
Frequency (Hz)
Magnitude (dB)
(b) y
FOM PROM: Range 24mm
Figure 4.6 Comparison of frequency responses of the FOM and the ROM at position (1): 24 (mm) sampling range.
0 1000 2000 3000 4000 5000
−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (dB)
(a) x
0 1000 2000 3000 4000 5000
−250
−200
−150
−100
−50 0
Frequency (Hz)
Magnitude (dB)
(b) y
FOM PROM: Range 24mm
Figure 4.7 Comparison of frequency responses of the FOM and the ROM at position (2): 24 (mm) sampling range.
0 0.005 0.01 0.015 0.02 0.025 0.03 0
0.5 1 1.5 2 2.5 3 3.5
Sampling Range (m)
Error (%)
Average Relative Error
Figure 4.8 Average relative error of 1st∼8th eigenvalues according to the sampling range.
0 0.005 0.01 0.015 0.02 0.025
0 0.5 1 1.5 2 2.5 3 3.5 4
Sampling Range (m)
Error (%)
Min, Max error Average error
Figure 4.9 Average relative error of 1st∼8th eigenvalues for random thickness
Table 4.2 Upper and lower bound of each interpolation cases Nominal: 7e-3 (m) (1) (2) (3) (4) # of samples
Linear - 5.5e-3 8.5e-3 - 16
Quadratic - 5.5e-3 8.5e-3 10.0e-3 81
Cubic 4.0e-3 5.5e-3 8.5e-3 10.0e-3 256
0 0.1
0.2 0.3
0.4 0.5
0.6 0.7
0.8 0
0.2
0.4
−0.1 0 0.1
z
x
y Sub. 1
µ1
Clamped
Sub. 2
µ2 Sub. 3
µ3 Sub. 4 µ4
Impulse Frequency
Response
Figure 4.10 Cantilever plate with 4 subdomains of under tip impluse load.
0 50 100 150 200 250 300 350 400 450 500
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(a) z
0 50 100 150 200 250 300 350 400 450 500
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(b) φ
x
0 50 100 150 200 250 300 350 400 450 500
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(c) φ
y
FOM PROM: 1st order
Figure 4.11 Comparison of frequency responses of the FOM and the ROM:
linear sampling.
0 50 100 150 200 250 300 350 400 450 500
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(a) z
0 50 100 150 200 250 300 350 400 450 500
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(b) φ
x
0 50 100 150 200 250 300 350 400 450 500
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(c) φ
y
FOM PROM: 2nd order
Figure 4.12 Comparison of frequency responses of the FOM and the ROM:
quadratic sampling.
0 50 100 150 200 250 300 350 400 450 500
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(a) z
0 50 100 150 200 250 300 350 400 450 500
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(b) φ
x
0 50 100 150 200 250 300 350 400 450 500
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(c) φ
y
FOM PROM: 3rd Poly. Lagrange Poly: 3rd
Figure 4.13 Comparison of frequency responses of the FOM, the ROM and Lagrange interpolation: cubic sampling.
0 100 200 300 400 500 0
0.5 1 1.5 2 2.5 3 3.5
Frequency (Hz)
Error (%)
1st Poly.
2nd Poly.
3rd Poly.
Figure 4.14 Comparison of the relative errors of eigenvalues using different polynomial order.
4 2 8 6
12 10 14
0
5
10
15
−4
−3
−2
−1 0 1 2 3 4
Rib 4 Impulse
Skin 8
Skin 7 Skin 1
Skin 2
Spar 3
Rib 6
Spar 5
Frequency Response
Clamped
Figure 4.15 Wing box model with 8 subdomains under tip impluse load.
0 20 40 60 80 100
−250
−200
−150
−100
Frequency (Hz)
Magnitude (dB)
0 20 40 60 80 100
−250
−200
−150
−100
Frequency (Hz)
Magnitude (dB)
0 20 40 60 80 100
−250
−200
−150
−100
Frequency (Hz)
Magnitude (dB)
0 20 40 60 80 100
−250
−200
−150
−100
Frequency (Hz)
Magnitude (dB)
0 20 40 60 80 100
−250
−200
−150
−100
Frequency (Hz)
Magnitude (dB)
0 20 40 60 80 100
−250
−200
−150
−100
Frequency (Hz)
Magnitude (dB)
FOM PROM
Figure 4.16 Comparison of frequency responses of the FOM and the ROM.
0 200 400 600 800 1000 0
0.5 1 1.5 2 2.5 3
Random Samples
Relative Error (%)
0 0.5 1 1.5 2
0 0.02 0.04 0.06 0.08 0.1
Relative Error (%)
Eigenvalue Error
PDF of Relative Error
min.
max.
mean median std.
0.09 1.95 0.67 0.62 0.29
Figure 4.17 Average of relative eigenvalue errors and probability density func- tion of 1,000 random samples.