**5. Optimization of a Vehicle Model**

**Figure 17.** Comparison of natural frequencies between original CCROM method and CCROM with reanalysis for the

Figure 17 compares the natural frequencies of the new (modified) design between the origi‐ nal CCROM (Craig-Bampton with Interface modes) method and the new approach where reanalysis is used in CCROM to approximate the interface modes. We observe that the natu‐ ral frequencies of the modified design are very different from those of the original design. Also, the accuracy of the proposed reanalysis method is excellent. The frequencies for the modified design calculated by the original CCROM and the proposed new approach are al‐ most identical. The percentage error of the new approach versus the original CCROM ap‐

**CCROM** 8 sec 61 sec 65 sec 3 sec 137 sec **New Approach** 7 sec **2 sec 0.3 sec** 2 sec 11 sec

**CCROM** 108 sec 282 sec 927 sec 10 sec 1327 sec **New Approach** 110 sec **16 sec 3 sec** 10 sec 139 sec

**Multiplication Other Cost Total Cost**

**Multiplication Other Cost Total Cost**

proach is less than 1% on average. The computation cost is summarized in Table 5.

**Constraint Modes**

**Constraint Modes**

car door example.

**Substructure 1:**

**Substructure 2:**

**CPU Time Normal**

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**CPU Time Normal**

**Modes**

**Modes**

**Table 5.** Summary of computational cost for the car door example.

A detailed optimization study is presented using a large-scale FE model of a vehicle. For simplicity, we call it "BETA" car model. It is composed of approximately 7.1 million DOFs and 1.1 million elements. Figure 18 shows all modeling details.

**Figure 18.** Details of "BETA" car model.

**Figure 21.** Description of the five design variables on the doors.

The optimization problem is stated as follows:

variables are schematically indicated in Figures 20 and 21.

tion problem is numerically very challenging because of

**2.** the computational cost of each dynamic analysis.

**1.** the many local optima and

Fifteen design variables are chosen; five structural elements of each door (thickness of door shell, front frame, rear frame, top panel, middle pipe), vertical stiffness of each of the four engine mounts, and vertical stiffness of each of the six exhaust system supports. All design

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The optimal value of each of the fifteen design variables is calculated in order to minimize the maximum response among the ten locations on the doors while the mass of the vehicle remains less or equal to the mass of the initial (nominal) vehicle. The response is calculated in the 100 Hz to 200 Hz frequency range and a 3% structural damping is used. The optimiza‐

The former was handled by using a hybrid optimization algorithm which first explores the entire design space using a Niching Genetic Algorithm (GA) [25] and then switches to a gra‐

**Figure 19.** Ten response locations on two front doors.

We form an optimization problem in terms of the maximum vibratory displacement at any location of the outer shell of the two front doors by minimizing the maximum displacement among ten locations of the two front doors (Figure 19) due to a hypothetical engine excita‐ tion in the vertical (up-down) direction. The engine is represented by a lumped mass con‐ nected rigidly to the engine mounts (Figure 20). The powertrain-exhaust model has about 1.3 million DOFs and is composed of 29 PSHELL components and 12 PSOLID components. There are also some RBE2 and PBUSH elements which are used as connectors. The maxi‐ mum displacement at each of the ten door locations is observed in the y direction (lateral direction – perpendicular to the door plane).

**Figure 20.** Description of the fifteen design variables.

**Figure 21.** Description of the five design variables on the doors.

**Figure 19.** Ten response locations on two front doors.

172 Advances in Vibration Engineering and Structural Dynamics

direction – perpendicular to the door plane).

**Figure 20.** Description of the fifteen design variables.

We form an optimization problem in terms of the maximum vibratory displacement at any location of the outer shell of the two front doors by minimizing the maximum displacement among ten locations of the two front doors (Figure 19) due to a hypothetical engine excita‐ tion in the vertical (up-down) direction. The engine is represented by a lumped mass con‐ nected rigidly to the engine mounts (Figure 20). The powertrain-exhaust model has about 1.3 million DOFs and is composed of 29 PSHELL components and 12 PSOLID components. There are also some RBE2 and PBUSH elements which are used as connectors. The maxi‐ mum displacement at each of the ten door locations is observed in the y direction (lateral Fifteen design variables are chosen; five structural elements of each door (thickness of door shell, front frame, rear frame, top panel, middle pipe), vertical stiffness of each of the four engine mounts, and vertical stiffness of each of the six exhaust system supports. All design variables are schematically indicated in Figures 20 and 21.

The optimization problem is stated as follows:

$$\begin{aligned}Find\\ X &= \left[X\_1 X\_2 \cdots X\_{1^\circ}\right] \\ \min\_{x} & \left[\max\_{i=1}^{10} (\text{Re} \, sp\_i)\right] \\ \text{such that} & : Mass \le Mass\_{\text{Nominal}} \\ where & \text{Re} \, sp = \left| \nu \left(f\right) \right| f \in \{100, 200\} \, Hz \\ & & \uparrow \\ & & \text{J} \, total \, door \, disulance \, net \end{aligned}$$

The optimal value of each of the fifteen design variables is calculated in order to minimize the maximum response among the ten locations on the doors while the mass of the vehicle remains less or equal to the mass of the initial (nominal) vehicle. The response is calculated in the 100 Hz to 200 Hz frequency range and a 3% structural damping is used. The optimiza‐

**1.** the many local optima and

**2.** the computational cost of each dynamic analysis.

tion problem is numerically very challenging because of

The former was handled by using a hybrid optimization algorithm which first explores the entire design space using a Niching Genetic Algorithm (GA) [25] and then switches to a gra‐ dient-based optimizer (fmincon in MATLAB) using the best estimate of the optimal point from the GA as initial point. This ensures a rapid convergence to the final optimum because although all GA optimizers can move quickly to the vicinity of the final optimum, they have a very slow convergence rate in pinpointing the final optimum.

FRF substructuring is used to assemble all components of the vehicle (body, doors, and en‐ gine-exhaust) into a small reduced-order model. This keeps the computational cost of each dynamic analysis low (4 minutes per analysis). A modal model is created only once for the body subsystem and then used to generate an FRF representation. This modal model does not change during the optimization because the chosen design variables are not associated with the body. However, the modal models of the doors change during the optimization. The final model for the entire vehicle is created by assembling the FRF models of each com‐ ponent. The FRF assembly operation is repeated during optimization because the FRF mod‐ els of the two doors keep changing.

The Niching GA optimizer maximizes a fitness function by modifying all design variables. A proper fitness function which minimizes the maximum response among the ten door loca‐ tions while satisfying the vehicle mass constraint is chosen as follows

**Figure 22.** Comparison of optimal and initial designs.

**Figure 23.** Vehicle local mode at 105 Hz indicating door deformation.

Table 6 compares the value of each design variable between the initial (nominal) and final optimal designs. It also indicates that all designed variables were allowed to vary within a lower and upper bound. The values of the five door design variables changed considerably between the initial and optimal designs. This is expected because the optimizer tried to sup‐ press the local door mode. The stiffness of the four engine mounts and the six exhaust sup‐ ports also changed. Although we intuitively expect the stiffness of the engine mounts to

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$$Fitness = \frac{\left[\max\{\text{Res } p\_i\}\right]\_{\text{Normal}}}{\left[\max\{\text{Res } p\_i\}\right]} \ast \left[1 + p \ast \min(c, \text{ 0})\right]$$

The ratio of the nominal maximum response over the actual maximum response is used so that the fitness value increases when the actual response is reduced. This ratio is multiplied by 1 <sup>+</sup> *<sup>p</sup>* \* min(*c*, 0) where p = 10 is a penalty value and*<sup>c</sup>* =1<sup>−</sup> *Mass MassNo*min*al* . Thus, c is positive if *Mass* is less than *MassNo*min*al* satisfying the constraint and the value of 1 + *p* \* min(*c*, 0) is equal to one.

Otherwise, c becomes negative if *Mass* is greater than*MassNo*min*al* and the term 1 + *p* \* min(*c*, 0) assumes a large negative value which reduces the fitness value considera‐ bly. As a result, the GA optimizer always satisfies the mass constraint while maximizing the value of the fitness function.

Figure 22 summarizes the optimization results by comparing the maximum door response between the optimal and initial designs. The optimizer determined that the maximum re‐ sponse occurs at location 9 (center of left front door of Figure 19) at approximately 105 Hz. Figure 23 shows that this represents a vehicle local mode involving motion of the doors on‐ ly. At the optimal design the maximum response was reduced from the initial 10-3 m to 0.47\*10-3 m (Table 6).

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**Figure 22.** Comparison of optimal and initial designs.

dient-based optimizer (fmincon in MATLAB) using the best estimate of the optimal point from the GA as initial point. This ensures a rapid convergence to the final optimum because although all GA optimizers can move quickly to the vicinity of the final optimum, they have

FRF substructuring is used to assemble all components of the vehicle (body, doors, and en‐ gine-exhaust) into a small reduced-order model. This keeps the computational cost of each dynamic analysis low (4 minutes per analysis). A modal model is created only once for the body subsystem and then used to generate an FRF representation. This modal model does not change during the optimization because the chosen design variables are not associated with the body. However, the modal models of the doors change during the optimization. The final model for the entire vehicle is created by assembling the FRF models of each com‐ ponent. The FRF assembly operation is repeated during optimization because the FRF mod‐

The Niching GA optimizer maximizes a fitness function by modifying all design variables. A proper fitness function which minimizes the maximum response among the ten door loca‐

) *No*min*al*

The ratio of the nominal maximum response over the actual maximum response is used so that the fitness value increases when the actual response is reduced. This ratio is multiplied

Otherwise, c becomes negative if *Mass* is greater than*MassNo*min*al* and the term 1 + *p* \* min(*c*, 0) assumes a large negative value which reduces the fitness value considera‐ bly. As a result, the GA optimizer always satisfies the mass constraint while maximizing the

Figure 22 summarizes the optimization results by comparing the maximum door response between the optimal and initial designs. The optimizer determined that the maximum re‐ sponse occurs at location 9 (center of left front door of Figure 19) at approximately 105 Hz. Figure 23 shows that this represents a vehicle local mode involving motion of the doors on‐ ly. At the optimal design the maximum response was reduced from the initial 10-3 m to

∗ 1 + *p* ∗min(*c*, 0)

*MassNo*min*al*

satisfying the constraint and the value of 1 + *p* \* min(*c*, 0) is

. Thus, c is positive

a very slow convergence rate in pinpointing the final optimum.

tions while satisfying the vehicle mass constraint is chosen as follows

max *i*=1 10

(Re*s pi*

(Re*s pi* )

max *i*=1 10

by 1 <sup>+</sup> *<sup>p</sup>* \* min(*c*, 0) where p = 10 is a penalty value and*<sup>c</sup>* =1<sup>−</sup> *Mass*

els of the two doors keep changing.

174 Advances in Vibration Engineering and Structural Dynamics

*Fitness* =

if *Mass* is less than *MassNo*min*al*

value of the fitness function.

0.47\*10-3 m (Table 6).

equal to one.

**Figure 23.** Vehicle local mode at 105 Hz indicating door deformation.

Table 6 compares the value of each design variable between the initial (nominal) and final optimal designs. It also indicates that all designed variables were allowed to vary within a lower and upper bound. The values of the five door design variables changed considerably between the initial and optimal designs. This is expected because the optimizer tried to sup‐ press the local door mode. The stiffness of the four engine mounts and the six exhaust sup‐ ports also changed. Although we intuitively expect the stiffness of the engine mounts to change but not the stiffness of the exhaust supports, this is not the case in this example. Ta‐ ble 6 also indicates that at the optimum we not only reduced the maximum response from 10-3 m to 0.47\*10-3 m but the vehicle mass was also reduced from the initial 55.12 units to the final 51.92 units.


**Figure 24.** Function evaluations of the Niching GA in the X1-X2-X3 space.

signs the Lazy Learning metamodeling technique will use downstream.

number of design variables and identify local and global optima.

**6. Conclusions and Future Work**

Considering that the computational cost for each function evaluation was 4 minutes, the to‐ tal computational time was (359 function evaluations) \* (4 minutes per evaluation) = 1436 minutes or 23.9 hours. This is acceptable considering the size and type of performed analy‐ sis. The computational cost, in terms of number of function evaluations, was kept low by coupling the Niching GA with a Lazy Learning metamodeling technique [26, 27]. The latter estimates the value of the fitness function from existing values at close by designs without calculating the actual response. It uses an error measure to figure out if the estimation is ac‐ curate. The error is small if enough previous designs, for which the fitness value was evalu‐ ated, are close to the new design. In this case, the metamodel estimates the current fitness value without running an actual dynamic response. If the error is large, an actual response is calculated and the fitness value of this new design is added to the "pool" of previous de‐

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Reduced-order models and reanalysis methodologies were presented for accurate and effi‐ cient vibration analysis of large-scale, finite element models, and for efficient design optimi‐ zation of structures for best vibratory response. The optimization is able to handle a large

For large FE models, it is common to solve for the system response through modal reduction in order to improve computational efficiency. An eigenanalysis is performed using the sys‐ tem stiffness and mass matrices and a modal model is formed which is then solved for the response. The computational cost can be also reduced using substructuring (or reduced-or‐ der modeling) methods. A modal reduction is applied to each substructure to obtain the component modes and the system level response is then obtained using component mode synthesis. In optimization of dynamic systems involving design changes (e.g. thicknesses,

**Table 6.** Summary of optimal design.

Figure 24 shows the actual function evaluations (design points where the vehicle dynamic response was calculated) in the X1-X2-X3 space and indicates the vicinity of the optimal de‐ sign point. The GA optimizer needed only 359 function evaluations and used a population size of 5\*(15+1) = 80 and a maximum of 10 generations. The population size and the number of allowed generations were kept at a minimum in order to locate the vicinity of the opti‐ mum quickly without "wasting" valuable computational effort.

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**Figure 24.** Function evaluations of the Niching GA in the X1-X2-X3 space.

change but not the stiffness of the exhaust supports, this is not the case in this example. Ta‐ ble 6 also indicates that at the optimum we not only reduced the maximum response from 10-3 m to 0.47\*10-3 m but the vehicle mass was also reduced from the initial 55.12 units to the

> **Upper Bound**

Figure 24 shows the actual function evaluations (design points where the vehicle dynamic response was calculated) in the X1-X2-X3 space and indicates the vicinity of the optimal de‐ sign point. The GA optimizer needed only 359 function evaluations and used a population size of 5\*(15+1) = 80 and a maximum of 10 generations. The population size and the number of allowed generations were kept at a minimum in order to locate the vicinity of the opti‐

mum quickly without "wasting" valuable computational effort.

**Nominal Design**

**Optimal Design**

final 51.92 units.

**Table 6.** Summary of optimal design.

**Thickness Lower**

176 Advances in Vibration Engineering and Structural Dynamics

**Bound**

X1 Door Shell 0.1 1 0.7 0.6638 X2 Front Frame 0.1 1 0.7 0.3084 X3 Rear Frame 0.1 1 0.7 0.2019 X4 Top Panel 0.1 2 0.7 0.2019 X5 Middel Pipe 0.5 4 2.4 1.3722 X6 Engine mount 19.5 370.5 195 110.6 X7 Engine mount 19.5 370.5 195 161.8 X8 Engine mount 19.5 370.5 195 108.8 X9 Engine mount 19.5 370.5 195 132.1 X10 Exhaust support 19.5 370.5 195 213.9 X11 Exhaust support 19.5 370.5 195 218.1 X12 Exhaust support 19.5 370.5 195 345.9 X13 Exhaust support 19.5 370.5 195 286.1 X14 Exhaust support 19.5 370.5 195 118.7 X15 Exhaust support 19.5 370.5 195 233.4 Max Resp. 1\*10-3 0.47\*10-3 Door Mass 55.12 51.92

**Design Variables**

> Considering that the computational cost for each function evaluation was 4 minutes, the to‐ tal computational time was (359 function evaluations) \* (4 minutes per evaluation) = 1436 minutes or 23.9 hours. This is acceptable considering the size and type of performed analy‐ sis. The computational cost, in terms of number of function evaluations, was kept low by coupling the Niching GA with a Lazy Learning metamodeling technique [26, 27]. The latter estimates the value of the fitness function from existing values at close by designs without calculating the actual response. It uses an error measure to figure out if the estimation is ac‐ curate. The error is small if enough previous designs, for which the fitness value was evalu‐ ated, are close to the new design. In this case, the metamodel estimates the current fitness value without running an actual dynamic response. If the error is large, an actual response is calculated and the fitness value of this new design is added to the "pool" of previous de‐ signs the Lazy Learning metamodeling technique will use downstream.

## **6. Conclusions and Future Work**

Reduced-order models and reanalysis methodologies were presented for accurate and effi‐ cient vibration analysis of large-scale, finite element models, and for efficient design optimi‐ zation of structures for best vibratory response. The optimization is able to handle a large number of design variables and identify local and global optima.

For large FE models, it is common to solve for the system response through modal reduction in order to improve computational efficiency. An eigenanalysis is performed using the sys‐ tem stiffness and mass matrices and a modal model is formed which is then solved for the response. The computational cost can be also reduced using substructuring (or reduced-or‐ der modeling) methods. A modal reduction is applied to each substructure to obtain the component modes and the system level response is then obtained using component mode synthesis. In optimization of dynamic systems involving design changes (e.g. thicknesses, material properties, etc) the FEA analysis must be repeated many times in order to obtain the optimum design. Also in probabilistic analysis where parameter uncertainties are present, the FEA analysis must be repeated for a large number of sample points. In such cas‐ es, the computational cost is very high, if not prohibitive.

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To drastically reduce the computational cost without compromising accuracy beyond an ac‐ ceptable level, we developed and used various reanalysis methods in conjunction with re‐ duced-order modeling, in optimization of vibratory systems. Reanalysis methods are intended to efficiently calculate the structural response of a modified structure without solv‐ ing the complete set of modified analysis equations. We presented a variety of reanalysis methods including the CDH/VAO method, the Combined Approximations (CA) and Modi‐ fied Combined Approximations (MCA) method, and the Parametric Reduced-Order Model‐ ing (PROM) method. Their advantages and limitations were fully described and demonstrated with practical examples.

Future work will concentrate on developing reanalysis methodologies for shape and topolo‐ gy optimization of vibratory systems and extend the presented work in optimization under uncertainty where efficient deterministic reanalysis methods will be combined with efficient probabilistic reanalysis methods.
