Chapter 5 Parametric Reduced Order Model with Substruc-
5.5 Numerical Results
wherefM LS is a simplified expression for the moving least square process as shown in Eqs. (4.32)∼(4.39). After finishing the recovery process in the sub- domain level, the assembly process for the full projection matrix is required, which is similar to the assembly of the mass and stiffness matrices. The rela- tions between the displacement vectors in different coordinates systems are expressed as
u1i u2i u3b
=T(µ)Φ3(µ)
ˆ u1p ˆ u2p
¯ u3p
, (5.48)
where
T(µ) =
Q1ip(µ1) 0 Ψ1ib(µ1) 0 Q2ip(µ2) Ψ2ib(µ2)
0 0 Ibb
. (5.49)
Consequently, the multiplication of two transformation matrix is the full projection matrix which is written as
P(µ) =T(µ)Φ3(µ)
=
Q1ip(µ1) 0 Ψ1ib(µ1)Φ3bq(µ) 0 Q2ip(µ2) Ψ2ib(µ2)Φ3bq(µ) 0 0 Φ3bq(µ)
. (5.50)
By multiplying P(µ) to the displacement obtained by solving the reduced system, the displacement of full system is recovered.
chapters, but the size and the number of subdomains are changed to deal with more number of design variables. In the component mode synthesis (CMS) process, the number of fixed interface normal mode is set to 10, and the number of total interface mode is 12 for both Craig-Bampton CMS and the developed method.
Figs. 5.2 and 5.2 represent the mean and min-max eigenvalue errors of 1,000 randomly generated thickness samples. As the order of interpolation increases, the relative error decreas to the values of Criag-Bamption com- ponent mode synthesis. The thicknesses of upper and lower bound of each cases were presented in table 5.1. Note that the number of samples increase algebraically different from in table 4.1. By using only cubic interpolation, the eigenvalue error is converge to that of CB CMS. Fig. 5.4 was obtained by changing the range of sample thicknesses maintaining the order of interpola- tion as cubic. If the range is narrow, almost exact agreement can be seen in (a). Even for (c) which has wide variation of the thicknesses, the error is still below 0.1 %.
For the structural optimization, dynamic load is applied to the tip and the loading profile is shown in Fig. 5.5. Time interval is [0:0.002:10] sec with 5,000 time steps. The objective function is the weight of the structure and other conditions are presented in table 5.2. In Fig. 5.6, the optimal thick- nesses of each method are presented: the FOM, modal reduction using 12 eigenmodes which solves the eigen-problem by using the Lanczos algorithm (more specifically, implicitly restarted Arnoldi method) and the parametric ROM (matrix interpolation with MLS recovery) with substructuring scheme.
The thicknesses of two ROMs show good agreements to that of the FOM.
The histories of the objective functions are shown in Fig. 5.7. In face, the objective functions of each models were already converged within 10∼15 iter- ations. After that, very slight decreases can be observed. By comparing each computational time, the efficiency of present method can be seen, which is 7
% and 33 % compared to the FOM and the modal reduction.
Example 2. Wing box structure
The second example is wing box model (Fig. 5.9) with ‘85’ design vari- ables and refined mesh configuration compared to the one shown in previous chapers. Total degrees of freedom is 72,438 with 12,560 elements and de- tails of condition for optimization were presented in talbe 5.3. Specific design variables were shown in Figs. 5.10 and 5.11.
The optimal thicknesses of spar and upper skin were given in Fig. 5.12 and that of rib and lower skin were shown 5.13. The optimal thicknesses of the present ROM shows better agreement than that of the modal reduction with the FOM. The histories of the objective functions (Fig. 5.14) are similar each other, which indicates the robustness of the present parametric ROM. Nev- ertheless, the present method is very efficient, even compared to the modal reduction as shown in Fig. 5.15; 2.1 % and 20.2 % efficient compared to the FOM and the modal reduction.
Example 3. High-fidelity F1 front wing
For the last example, high-fidelity front wing structure of fomular-1 machine
Total degrees of freedom of the full model is 749,082. However, to optimize the size of structural components, the symmetric boundary condition is ap- plied, which results in the half model with 375,588 degrees of freedom in Fig.
5.17. The number of design variables is ‘96’. The material properties were assumed to be carbon composites: elastic modulus E = 70e9 Pa, Poisson’s ratio ν = 0.25, density ρ = 1600 Kg/m3 and the initial thicknesses are 8 mm for overall domains. The external loading profile is denoted in Fig. 5.18.
In Fig. 5.17, vertical down forces are applied to the red points, negative x- directional forces are applied to cyan points and negative y-directional forces are applied to yellow point. In fact, all nodes are under external forces which are occured by aerodynamics, and to do so, the CFD-based fluid simula- tion should be performed first. However, we simplified the external forces to the nodal loads to avoid a complicated CFD computations since we can still observe the efficiency of the present parametric ROM under the simpli- fied loading conditions. The complicated loadings also can be applied to the present method without any difficulties.
Fig. 5.19 and 5.20 show the optimal thicknesses of two different ROMs.
Both methods have good agreements with each other. There are slight dif- ferences for the optimal thicknesses of subdomain 42∼96. However the ten- dencies are almost the same. In Fig. 5.21, the histories of objective function were compared, which is also similar to each other. Compared to the result of the wing box problem in Fig. 5.14, the result of present F1 model con- verged faster than that of the wing box model. Because, whereas the F1 model has distributed loading point, the wing box model has only 1 loading point. Therefore, the design variables of the F1 are more sensitive than that
of the wing box model. The computation time of the parametric ROM is 22.7% compared to the modal reduction method. Considering the efficiency increases as the degrees of freedom of FOM increase, estimated efficiency compared to the FOM could be lower than 2.1 %.
Table 5.1 Upper and lower bound of each interpolation cases
(m) (1) (2) (3) (4) # of samples
Linear - 10e-3 15e-3 - 16
Quadratic - 10e-3 15e-3 20e-3 24 Cubic 5e-3 10e-3 15e-3 20e-3 32
Table 5.2 Problem condition of rib-skin-spare structure
Weight (Kg) µlb (m) µub (m) |umax|(m) ω1 (Hz) |σmax|(Pa)
196.83 5e-3 20e-3 10e-3 12 3e9
0 0.5
1 1.5
2 2.5
3 0
0.5 0
0.1 0.2
5 8 4 7
2 1
Dynamic Load f(t)
Clamped 3
6
Figure 5.1 Rib-skin-spar structure with 8 subdomains under tip dynamic load.
0 2 4 6 8 10 12 14 16 18 20 10−6
10−4 10−2 100 102
(a)
Mode Number
Relative Error (%)
0 2 4 6 8 10 12 14 16 18 20
10−6 10−4 10−2 100 102
(b)
Mode Number
Relative Error (%)
0 2 4 6 8 10 12 14 16 18 20
10−6 10−4 10−2 100 102
(c)
Mode Number
Relative Error (%)
PROM w Substr. CB CMS
Figure 5.2 Comparison of mean of eigenvalue errors for 1,000 random sam- ples: The Craig-Bampton component mode systhesis and the (a) linear, (b)
0 2 4 6 8 10 12 14 16 18 20 10−8
10−6 10−4 10−2 100 102
(a)
Mode Number
Relative Error (%)
0 2 4 6 8 10 12 14 16 18 20
10−8 10−6 10−4 10−2 100 102
(b)
Mode Number
Relative Error (%)
0 2 4 6 8 10 12 14 16 18 20
10−8 10−6 10−4 10−2 100 102
(c)
Mode Number
Relative Error (%)
PROM w Substr. CB CMS
Figure 5.3 Comparison of min-max of eigenvalue errors for 1,000 random samples: The Craig-Bampton component mode systhesis and the (a) linear, (b) quadratic and (c) cubic interpolated ROM.
0 2 4 6 8 10 12 14 16 18 20 10−6
10−4 10−2 100
(a)
Mode Number
Relative Error (%)
0 2 4 6 8 10 12 14 16 18 20
10−6 10−4 10−2 100
(b)
Mode Number
Relative Error (%)
0 2 4 6 8 10 12 14 16 18 20
10−6 10−4 10−2 100
(c)
Mode Number
Relative Error (%)
PROM w Substr. CB CMS
Figure 5.4 Comparison of mean of eigenvalue errors for 1,000 random samples:
The Craig-Bampton component mode systhesis and the cubic interpolated
0 2 4 6 8 10 0
200 400 600 800 1000 1200
time(sec)
F (N)
F(t)
Figure 5.5 Dynamic step loading profile.
0 1 2 3 4 5 6 7 8 9
0.005 0.01 0.015 0.02
Design Variables
Thickness (m)
FOM
Modal Reduc. (Lanczos) PROM w Substr.
Figure 5.6 Comparison of optimal thicknesses of the FOM and ROMs.
0 5 10 15 20 25 30 0.75
0.8 0.85 0.9 0.95 1
Iterations
|Obj. function|
FOM
Modal Reduc. (Lanczos) PROM w Substr.
Figure 5.7 Comparison of objective function histories.
1 2 3
0 500 1000 1500 2000
1891.79
404.64
133.83
time (sec)
1. FOM
2. Modal Reduc. (Lanczos) 3. PROM w Substr.
Table 5.3 Problem condition of wing box model Weight (Kg) µlb (m) µub (m) |umax|(m) ω1 (Hz)
8320.5 5e-3 20e-3 5e-3 3.5
2 4
6 8
10
12 14
0 5
10 15
−0.5 0 0.5
Dynamic load f(t)
Clamped
Figure 5.9 Wing box model with 85 subdomains under tip dynamic load.
2 4
6 8
10 12
14
0
5
10
15
−0.5 0 0.5
1 2
26 6 3
7 27
30 11 8 28
31 4
12 34 16 9
13 32 29
35 5
17 38 14 33
36 10
18
21 39 22 15
37 19
41 23 45 20 40
44 24 43 25 42
Figure 5.10 Design variables of rib and spar.
2 4 6 8 10 12 14
0
5
10
15
−4
−2 0 2 4
46
47 49 48
54
55 51 50
56
57 62
63 53
59 52
58
64
65 61 60
70 66
67
71 72
73 69 68
74
75
79 78 76
77
81 80
83 82
85 84
Figure 5.11 Design variables of upper and lower skins.
0 5 10 15 20 25 0
0.005 0.01 0.015 0.02
(a) Spar
Design Variables
Thickness (m)
0 5 10 15 20
0 0.005 0.01 0.015 0.02
(b) Upper Skin
Design Variables
Thickness (m)
FOM
Modal Reduc. (Lanczos) PROM w Substr.
Figure 5.12 Comparison of optimal thicknesses of the FOM and ROMs of
0 5 10 15 20 0
0.005 0.01 0.015 0.02
(c) Rib
Design Variables
Thickness (m)
0 5 10 15 20
0 0.005 0.01 0.015 0.02
(d) Lower Skin
Design Variables
Thickness (m)
FOM
Modal Reduc. (Lanczos) PROM w Substr.
Figure 5.13 Comparison of optimal thicknesses of the FOM and ROMs of rib and lower skin
0 5 10 15 20 25 30 0.4
0.5 0.6 0.7 0.8 0.9 1
Iterations
|Obj. function|
FOM
Modal Reduc. (Lanczos) PROM w Substr.
Figure 5.14 Comparison of objective function histories.
1 2 3
0 50 100 150 200 250
194.98
20.23
4.08
Time (h)
1. FOM
2. Modal Reduc. (Lanczos) 3. PROM w Substr.
Table 5.4 Problem condition of high-fidelity F1 front wing model Weight (Kg) µlb (m) µub (m) |umax|(m) ω1 (Hz)
19.24 5e-3 10e-3 5e-3 24
Figure 5.16 Configureation of high-fidelity F1 front wing structure.
Figure 5.17 Half of F1 front wing with 96 subdomains under multiple dynamic loads
0 2 4 6 8 10 0
10 20 30 40 50 60 70 80
time (sec)
F (N)
z−dir. (red points) x−dir. (cyan points) y−dir. (yellow point)
Figure 5.18 Dynamic loads applied to each points
0 5 10 15 20 0
0.002 0.004 0.006 0.008 0.01
(a) Sub. # 1~20
Design Variables
Thickness (m)
20 25 30 35 40
0 0.002 0.004 0.006 0.008 0.01
(b) Sub. # 21~41
Design Variables
Thickness (m)
Modal Reduc. (Lanczos) PROM w Substr.
45 50 55 60 65 70 0
0.002 0.004 0.006 0.008 0.01
(c) Sub. # 42~69
Design Variables
Thickness (m)
70 75 80 85 90 95
0 0.002 0.004 0.006 0.008 0.01
(d) Sub. # 70~96
Design Variables
Thickness (m)
Modal Reduc. (Lanczos) PROM w Substr.
Figure 5.20 Comparison of optimal thicknesses of the FOM and ROMs of subdomain # 42∼96
0 2 4 6 8 10 12 14 16 18 0.7
0.75 0.8 0.85 0.9 0.95 1 1.05
Iterations
|Obj. function|
Modal Reduc. (Lanczos) PROM w Substr.
Figure 5.21 Comparison of objective function histories.
1 2
0 10 20 30 40 50 60 70 80
60.48
13.70
Time (h)
1. Modal Reduc. (Lanczos) 2. PROM w Substr.