Chapter 6 Stochastic Dynamic Analysis with Uncertain Pa-
6.4 Numerical Results
Example 1. Cantilever plate
First of all, rectangular cantilever plate was investigated as shown in Fig. 6.1 with 4 uncertain parameters. The geometries and material properties are the same to that of the cantilever plate in Chap. 4. The thickness of the plate is 7 mm and the uncertain parameter is the elastic modulus of each subdomains.
The norminal modulus is set to be 73.1e9 Pa and 10 % variation is assumed in this example.
From the beta distribution, 1,000 random vectors of elastic modulus were generated as shown in Fig. 6.2. The frequency response analysis was executed in the range of [0:0.1:100] Hz. For the ROM, 21 snapshots of frequency re- sponse were taken in [0:5:100] Hz. The coefficients of beta distribution are simply assumed to be q = r = 3. In this example, the ROM in Chap 4. is considered. Since the substructuring method is efficient with many numbers of parameters.
Figs. 6.3, 6.4 and 6.5 represent average mean, maximum and minimum responses of the FOM and the linearly interpolated ROM. For the mean values, there is no difference between the FOM and the ROM. For the max and min, overall distributions of the ROM have a good agreement to that of the FOM except the peak points. In fact, the peak value itself does not have significant meaning. And the ROM can express the migration of eigenvalues according to the random samples. The computation times were presented in
Example 2. Rib-skin-spar structure
The second example is the rib-skin-spar structure presented in Fig. 6.6. The uncertain parameters are the elastic modulus of each subdomain, totally 8 parameters. The geometris and material properties are the same to that of the structure in Chap. 5. The thickness, however, is constant value 15 mm. The norminal elastic modulus is 72e9 Pa and the variation of uncertain parameter is also assumed as 10 %. The frequency range is [0:0.2:100] Hz and also 20 snapshots were taken in [0:5:100] Hz. THe coefficients of beta distribution are the same to that of example 1.
Both of the parametric ROMs in Chap. 4 and 5 were investigated. The method in Chap. 4 is presented as ‘PROM: linear interp.’ and the one in Chap. 5 is ‘PROM w Substr’. The ‘PROM w Substr’ method interpolated by using cubic polynomial. As shown in table 6.1, the construction time of both ROM is very different. For the first method, totally 28 = 256 numbers of full model computations were performed to construct the ROM. However, the second method have 32 numbers of subdomain computations. Therefore, almost no time takes to construct the ROM. By contrast, the analysis time of the first method is faster than that of the second method. Due to the subdomain synthesis and the interface reduction, the analysis time increase compared to the first method. The total time of the first and the second method are 2.1 % and 0.9 % compared to the full model, which indicates that both methods are very efficient.
In Fig. 6.7, average mean response looks similar with each other. However, the degree of freedom of y-directional rotation of the method in Chap. 4 shows
slight differences. This becomes more serious for the maximun and minimum responses as shown in Figs. 6.8 and 6.9. Therefore, we can conclude that the performance of the parametric ROM with substructuring scheme is better than that of the ROM in Chap. 4, especially when the structure has large numbers of parameters.
Table 6.1 Computation time of the FOM and ROMs of cantilever plate
Time (sec) ROM FOM
Construction 1.40 Analysis 23.47
Total 24.87 622.90
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
Clamped
Impulse Frequency
Response Sub. 1
E1 Sub. 2
E2 Sub. 3
E3 Sub. 4 E4
Figure 6.1 Cantilever plate with 4 uncertain parameters.
66 68 70 72 74 76 78 80 0
0.2 0.4 0.6 0.8 1 1.2 1.4x 10−3
E (GPa) (a)
E1
66 68 70 72 74 76 78 80
0 0.2 0.4 0.6 0.8 1 1.2 1.4x 10−3
E (GPa) (b)
E2
66 68 70 72 74 76 78 80
0 0.2 0.4 0.6 0.8 1 1.2 1.4x 10−3
E (GPa) (c)
E3
66 68 70 72 74 76 78 80
0 0.2 0.4 0.6 0.8 1 1.2 1.4x 10−3
E (GPa) (d)
E4
Figure 6.2 PDF of elatic modulus of each substructures.
0 20 40 60 80 100
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(a) z
0 20 40 60 80 100
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(b) φ
x
0 20 40 60 80 100
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(c) φ
y
FOM PROM: 1st order
Figure 6.3 Average mean frequency responses of the FOM and the ROM
0 20 40 60 80 100
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(a) z
0 20 40 60 80 100
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(b) φx
0 20 40 60 80 100
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(c) φy
FOM PROM: 1st order
Figure 6.4 Average maximum frequency responses of the FOM and the ROM
0 20 40 60 80 100
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(a) z
0 20 40 60 80 100
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(b) φ
x
0 20 40 60 80 100
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(c) φ
y
FOM PROM: 1st order
Figure 6.5 Average minimum frequency responses of the FOM and the ROM
Table 6.2 Computation time of the FOM and ROMs of rib-skin-spar structure Time (sec) ROM w/o Substr. ROM w. Substr. FOM
Construction 614.6 2.3
Analysis 107.6 287.3
Total 722.2 289.3 33859.90
0 0.5
1 1.5
2 2.5
3 0
0.5 0
0.1 0.2
5 8 4 7
2 1
Clamped 3
6 Impulse
Frequency Response
Figure 6.6 Rib-skin-spar structure with 8 uncertain parameters
0 10 20 30 40 50 60 70 80 90 100
−200
−150
−100
−50
Frequency (Hz)
Magnitude (dB)
(a) z
0 10 20 30 40 50 60 70 80 90 100
−200
−150
−100
−50
Frequency (Hz)
Magnitude (dB)
(b) φ
x
0 10 20 30 40 50 60 70 80 90 100
−200
−150
−100
−50
Frequency (Hz)
Magnitude (dB)
(c) φ
y
FOM PROM w Substr. PROM: linear interp.
Figure 6.7 Average mean frequency responses of the FOM and ROMs
0 10 20 30 40 50 60 70 80 90 100
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(a) z
0 10 20 30 40 50 60 70 80 90 100
−150
−100
−50 0 50
Frequency (Hz)
Magnitude (dB)
(b) φx
0 10 20 30 40 50 60 70 80 90 100
−200
−100 0 100
Frequency (Hz)
Magnitude (dB)
(c) φy
FOM PROM w Substr. PROM: linear interp.
Figure 6.8 Average maximum frequency responses of the FOM and ROMs
0 10 20 30 40 50 60 70 80 90 100
−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (dB)
(a) z
0 10 20 30 40 50 60 70 80 90 100
−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (dB)
(b) φx
0 10 20 30 40 50 60 70 80 90 100
−300
−250
−200
−150
−100
Frequency (Hz)
Magnitude (dB)
(c) φy
FOM PROM w Substr. PROM: linear interp.
Figure 6.9 Average minimum frequency responses of the FOM and ROMs
Chapter 7
Conclusions
In this dissertation, parametric reduced order models for comprising the dy- namic characteristics and the change of parameters were developed within the finite element framwork. Based on the characteristics of the proper orthog- onal decomposition, enhanced reduced basis method was developed to treat multiple loading conditions. By calculating the mode of multiple loads, the efficiency of constructing reduced basis was increased. In addition, efficient design optimization strategy for dynamic response was suggested using re- duced equivalent static load calculated by using the global proper orthogonal decomposition.
To obain real-time, on-line parametric reduced order model, projection- transformation-recovery procedure was developed by employing the global proper orthogonal method for computing transformation matrix and by using the moving least square approximation with recovery process. The accuracy and robustness of the proposed method were decomstrated by the frequency response analysis of various examples. Whereas the eigenvalues are interpo-
lated well, the eigenvectors consisting the basis of reduced space cannot be accuratly interpolated by using conventional Lagrange interpolation function.
Therefore, moving least square method was employed to calculate accurate projection matrix. Parametric studies provided the addmissible variation of parameters to employ the proposed method within certain error bounds.
The computation on the off-line stage was also reduced by introducing substructuring scheme to the parametric reduced order model. For the struc- tural design optimization, computational time consumed in approximating the global response according to the change of parameters is also significant.
The substructuring scheme facilitated to calculate the global response in a subdomain level. Thus, both on-line and off-line calculations were reduced, which results in the fast computation of large-scale structures contains many design variables up to hundreds level. Considering the computations were ex- ecuted in the desktop PC, extreme-scale problems could be solved by using super computing system.
Based on the analysis and design optimization of high-fidelity model for dynamic response performed in this dissertation, it is hoped that the present optimization strategy and parametric reduced order model can be further employed to other structural applications.
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국문 요약
본 논문에서는 동적 특성과 파라미터의 변화를 동시에 고려하는 유한요 소 기반의 파라메트릭 축소 기법을 개발하였다. 기존의 축소 기법은 동적 특성이나 파라미터 변화를 개별적으로 축소하기 때문에, 동적 시스템에서 파라미터가 변하면 축소 시스템을 재구성해야 하며,이경우 계산 효율성이 낮아지는 문제가 발생한다. 이를 해결하기 위해서 적합 직교 분해 기반의 파라메트릭 축소기법을 제안하였다.
먼저,적합 직교 분해의 특성에 기반하여,하중의축소를 통해 다중하중 문제를 효율적으로 접근할 수 있는 축소 기저법을 제안하였다. 다중 하중 문제의 경우 기존의 방법으로는 하중이 변할 때 축소 기저를 재구축해야 하지만, 본 연구에서는 전역 적합 직교 분해 기법을 이용하여 축소 기저를 재구축하지 않는 방법을 고안하였다.이 방법은등가정하중을 이용한최적 설계 기법과 결합하여, 동적 시스템의 최적 설계 시, 계산 효율성이 증가 함을 확인하였다. 또한, 파라미터 변화를 실시간으로 고려하기 위해서, 투 영-좌표변환-보간-회복으로 구성되는 보간 기반의 축소기법을 제안하였다.
이 방법은 이동 최소 자승법과 결합하여, 기존의 라그랑지 보간법에 비해 월등히 정확하게 축소 시스템을 전체 시스템으로 회복시킬 수 있는 것을 확인하였다.
한편,파라메트릭 축소 모델을 대형 동적 시스템의 최적 설계 문제에 적 용하기위해서는,민감도 계산을 비롯한 최적설계반복연산시간의축소가 필요할 뿐만 아니라, 반복 연산 이전에 근사화된 전역 반응면의 탐색 시간 또한 줄어들어야 한다. 따라서 기존의 파라메트릭 축소 모델과 부구조화