**4. Reanalysis Methods in Dynamic Analysis and Optimization**

The reanalysis methods of Section 3 can be used in different dynamic analyses such as modal or direct frequency response and free or forced vibration in time domain. Depend‐ ing on the problem and the type of analysis, a particular reanalysis method may be prefer‐ red considering how many times it will be performed and how many design parameters will be allowed to change. This section demonstrates the computational efficiency and ac‐ curacy of reanalysis methods in dynamic analysis and optimization. It also introduces a new reanalysis method in Craig-Bampton substructuring with interface modes which is very useful for problems with many interface DOFs where FRF substructuring is not prac‐ tically applicable.

## **4.1. Integration of MCA Method in Optimization**

Each group *T <sup>i</sup>*

PROM methods.

Basis Vector Variable basis *R*/*T* .

Reanalysis Cost Moderate

Size proportional to *n* and *s*.

Relatively small size of *R*/*T* .

Best Application Small number of reanalyses compared to the number of design parameters.

gradient-based optimization.

**Table 1.** Comparison of the CA/MCA and PROM methods.

Must recalculate *R*/*T* at every new design.

Evaluation of few design alternatives and

contains roughly *n* / *k* original modes *Φ*<sup>0</sup>

reduces the cost considerably with minimal loss of accuracy.

**3.5. Comparison of CA/MCA and PROM Methods**

156 Advances in Vibration Engineering and Structural Dynamics

improved modes. The eigenvectors of the new design are calculated using *T <sup>i</sup>*

Equations (62) to (64). The process is repeated *k* times using a modal basis that is 1 / *k* of the size of the original modal basis. All *k* groups of eigenvectors are then collected together to calculate the frequency response of the new design. As demonstrated in Section 4.3.2, this

As we have discussed, a large overhead cost which is proportional to the number of design parameters is required before the PROM reanalysis is carried out. However, the CA/MCA method does not require this overhead cost because it does not need the basis *P* of Equation (49) (see Section 3.2). Therefore, the CA/MCA method is more suitable, when the number of reanalyses is comparable to the number of design parameters. This is usually true in gradi‐ ent-based design optimization. The CA and MCA methods can become expensive however, when many reanalyses are needed because, for each reanalysis, they require a new basis *R* or *T* (see Equations 56and 61, respectively) and new condensed mass and stiffness matrices in Equations (57) and (62). This is the case in gradient-free optimization problems employ‐ ing a Genetic Algorithm for example, and in simulation-based probabilistic analysis prob‐ lems employing the Monte-Carlo method. For these problems, the PROM method is more suitable because the subspace basis *P* does not change for every new design point. Table 1 summarizes the main characteristics, advantages and application areas of the CA/MCA and

**CA/MCA Method PROM Method**

parameters *m*.

Constant basis *P*.

Size proportional to *n* and *m*.

Very large number of reanalyses. Gradient-free optimization (e.g. genetic algorithms) and probabilistic analysis.

Cost proportional to the number of design

High if no parametric relationship exists due to the condensation of large size and dense *P*. Low if a parametric relationship exists.

Overhead Cost None Required cost to construct *P*.

*<sup>i</sup>* from*Φ*0, and their corresponding

instead of T in

We have mentioned that the MCA method provides a good balance between accuracy and efficiency for problems that require a moderate number of reanalyses, as in gradient-based optimization. For problems where a large number of reanalyses is necessary, as in probabil‐ istic analysis and gradient-free (e.g. genetic algorithms) optimization, a combination of the MCA and PROM methods is more suitable.

Figure 6 shows a flowchart of the optimization process for modal frequency response prob‐ lems. The DMAP (Direct Matrix Abstraction Program) capabilities in NASTRAN have been used to integrate the MCA method and the NASTRAN modal dynamic response and opti‐ mization (SOL 200). The highlighted boxes indicate modifications to the NASTRAN opti‐ mizer. Starting from the original design, the code first calculates the design parameter sensitivities in order to establish a local search direction and determine an improved design along the local direction. At the improved design, an eigen-solution is obtained to calculate a modal model and the corresponding modal response. The dynamic response at certain physical DOFs is then recovered from the modal response. At this point, a convergence check is performed to decide if the optimal design is obtained. If not, further iterations are needed and the above procedure is repeated. Many iterations are usually needed for practi‐ cal problems to obtain the final optimal design. Section 4.3.2 demonstrates how this process was used to optimize the vibro-acoustic behavior of a 65,000 DOF, finite-element model of a truck. Using the MCA method, the computational cost of the entire optimization process was reduced in half compared with the existing NASTRAN approach.

As for a stand alone modal frequency response, the eigen-solution accounts for a large part of the overall optimization cost for vibratory problems where a modal model is used. A re‐ analysis method can be inserted into the procedure as shown in Figure 6 to provide an ap‐ proximate eigen-solution saving therefore, substantial computational cost. Other reanalysis methods such as the CDH/VAO, CA or PROM can also be used depending on the number of design variables and the number of expected iterations.

*T* = *Φ<sup>o</sup> T*0,*<sup>s</sup> T*1,*<sup>s</sup>* ⋯ *Tm*,*<sup>s</sup>* (68)

*<sup>p</sup>* can be subsequently used in a modal dynamic response solution. Only

step 4 is repeated in reanalysis. The computational savings can be substantial especially for

Vibration and Optimization Analysis of Large-Scale Structures using Reduced-Order Models and Reanalysis Methods

The pickup truck vehicle model with 65,000 DOFs of Figure 7 is used in this section to dem‐ onstrate the advantages of the combined MCA and PROM method in optimizing the vibroacoustic response of a vehicle. The model has 78 components and roughly 11,000 nodes and elements. The example is performed on a SUN ULTRA workstation using NASTRAN v2001.

The sound pressure level at the driver's ear location is calculated using a vibro-acoustic analysis. The structural forced vibration response due to unit harmonic forces in x, y, and z directions at the engine mount locations, is coupled with an interior acoustic analysis. The first and second eigen-frequencies of the acoustic volume inside the cabin are 95.9 Hz and 128.3 Hz. The sound pressure level is calculated in the 80 to 140 Hz frequency range. The structure and fluid domains are coupled through boundary conditions ensuring continuity

**4.3. Combined MCA and PROM Methods: Vibro-Acoustic Response of a Vehicle**

The MCA and PROM methods have been implemented in NASTRAN DMAP.

*<sup>p</sup>* using the subspace projection procedure of

http://dx.doi.org/10.5772/51402

159

where *m* is the total number of parameters. **•** Obtain the approximate mode shapes *Φ***˜**

problems where many reanalyses are needed.

The modal basis *Φ***˜**

**Figure 7.** FE model of a pickup truck.

Equations (50) through (52) where *T* is used instead of*P*.

**Figure 6.** Flowchart for mca-enhanced nastran optimization.

#### **4.2. Integration of MCA and PROM Methods**

The PROM method requires exact calculation of the mode shapes for all designs correspond‐ ing to the corner points of the parameter space in order to calculate the subspace basis *P*of Equation (49). The required computational effort can be prohibitive for a large number of parameters (optimization design variables). This effort can be reduced substantially if the modes of each corner point are approximated by the MCA method. In this case, an exact ei‐ gen-solution is required only for the baseline design. The following steps describe an algo‐ rithm to integrate the MCA and PROM methods:


$$\begin{aligned} \mathbf{T}\_{i,1} &= \mathbf{K}\_i^{-1} (\mathbf{M}\_i \boldsymbol{\Phi}\_o) \\ \mathbf{T}\_{i,j+1} &= \mathbf{K}\_i^{-1} (\mathbf{M}\_i \mathbf{T}\_{i,j}) \qquad j = \text{2, 3, } \cdots \text{, s} \end{aligned} \tag{67}$$

**•** Form the subspace basis **T** as

Vibration and Optimization Analysis of Large-Scale Structures using Reduced-Order Models and Reanalysis Methods http://dx.doi.org/10.5772/51402 159

$$T = \begin{bmatrix} \boldsymbol{\Phi}\_o & T\_{0,s} & T\_{1,s} & \cdots & T\_{m,s} \end{bmatrix} \tag{68}$$

where *m* is the total number of parameters.

**•** Obtain the approximate mode shapes *Φ***˜** *<sup>p</sup>* using the subspace projection procedure of Equations (50) through (52) where *T* is used instead of*P*.

The modal basis *Φ***˜** *<sup>p</sup>* can be subsequently used in a modal dynamic response solution. Only step 4 is repeated in reanalysis. The computational savings can be substantial especially for problems where many reanalyses are needed.

#### **4.3. Combined MCA and PROM Methods: Vibro-Acoustic Response of a Vehicle**

The pickup truck vehicle model with 65,000 DOFs of Figure 7 is used in this section to dem‐ onstrate the advantages of the combined MCA and PROM method in optimizing the vibroacoustic response of a vehicle. The model has 78 components and roughly 11,000 nodes and elements. The example is performed on a SUN ULTRA workstation using NASTRAN v2001. The MCA and PROM methods have been implemented in NASTRAN DMAP.

**Figure 7.** FE model of a pickup truck.

**Figure 6.** Flowchart for mca-enhanced nastran optimization.

158 Advances in Vibration Engineering and Structural Dynamics

**4.2. Integration of MCA and PROM Methods**

rithm to integrate the MCA and PROM methods:

**•** Use the MCA method at design point **p**<sup>i</sup>

**•** Form the subspace basis **T** as

lower limit, to obtain the baseline mode shapes*Φ*0.

corner point using the following recursive process

*Ti*,1 = *K<sup>i</sup>* −1 (*M<sup>i</sup> Φo*)

*Ti*, *<sup>j</sup>*+1 = *K<sup>i</sup>*

−1 (*M<sup>i</sup> Ti*, *<sup>j</sup>*

The PROM method requires exact calculation of the mode shapes for all designs correspond‐ ing to the corner points of the parameter space in order to calculate the subspace basis *P*of Equation (49). The required computational effort can be prohibitive for a large number of parameters (optimization design variables). This effort can be reduced substantially if the modes of each corner point are approximated by the MCA method. In this case, an exact ei‐ gen-solution is required only for the baseline design. The following steps describe an algo‐

**•** Perform an exact eigen-analysis at the baseline design point **p**0 all parameters are at their

*i th* parameter which is set at its upper limit. Obtain approximate mode shapes for the *i th*

) *j* =2, 3, ⋯, *s*

with all parameters at their low limit except the

(67)

The sound pressure level at the driver's ear location is calculated using a vibro-acoustic analysis. The structural forced vibration response due to unit harmonic forces in x, y, and z directions at the engine mount locations, is coupled with an interior acoustic analysis. The first and second eigen-frequencies of the acoustic volume inside the cabin are 95.9 Hz and 128.3 Hz. The sound pressure level is calculated in the 80 to 140 Hz frequency range. The structure and fluid domains are coupled through boundary conditions ensuring continuity of vibratory displacement and acoustic pressure. A finite-element formulation of the cou‐ pled undamped problem yields the following system equations of motion [24].

$$
\begin{aligned}
\begin{bmatrix}
\begin{bmatrix}
\mathbf{K}\_{S} & -\mathbf{H}\_{SF} \\
\mathbf{0} & \mathbf{K}\_{F}
\end{bmatrix} - j o^{2}
\begin{bmatrix}
\mathbf{M}\_{S} & \mathbf{0} \\
\rho\_{0} c\_{0}^{2} \mathbf{H}\_{SF}^{T} & \mathbf{M}\_{F}
\end{bmatrix}
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
\mathbf{d}\_{s} \\
\mathbf{p}\_{F}
\end{bmatrix} = \begin{bmatrix}
\mathbf{f}\_{b} \\
\mathbf{f}\_{q}
\end{bmatrix}
\end{aligned}
\tag{69}
$$

using NASTRAN. The computational cost to construct the reduced basis (*P*in PROM and *T* in PROM+MCA) is compared in Table 2. The total cost was reduced from 1080 seconds to 330 seconds. The computational saving is more significant if the number of design parame‐

Vibration and Optimization Analysis of Large-Scale Structures using Reduced-Order Models and Reanalysis Methods

**Figure 8.** Comparison of sound pressure at driver's ear between combined MCA and PROM method and NASTRAN.

PROM 180 sec 180\*5=900 sec 1080 sec PROM+MCA 180 sec 30\*5=150 sec 330 sec

The goal here is to minimize the sound pressure at the driver's ear. A total of 41 design pa‐ rameters are used representing the thickness of all vehicle components modeled with plate elements. All thicknesses are allowed to change by 100% from their baseline values. Table 3 describes all design parameters. At the initial point of the optimization process, all parame‐

**Solving for mode shapes at 5 corner design points**

**Total Cost**

http://dx.doi.org/10.5772/51402

161

**Method Solving for mode**

**Table 2.** CPU time to construct reduced basis.

*4.3.2. Optimization using MCA Method*

ters are at their low bound.

**shape Φ0 at baseline design**

ter increases.

where the vibratory displacement *d<sup>S</sup>* and the acoustic pressure *p<sup>F</sup>* are the primary variables. The finite-element representation of the two domains consists of stiffness and mass matrix pairs *K<sup>S</sup>* , *MS* and *K<sup>F</sup>* , *M<sup>F</sup>* , respectively. The air density and wave speed are *ρ*0 and*c*0. The right hand side of Equation (69) denotes the external forces.

The spatial coupling matrix *HSF* indicates coupling between the two domains which is usu‐ ally referred to as "two-way coupling." Due to this coupling, the combined structural-acous‐ tic system of equations is not symmetric. If the acoustic effect on the structural response is small, the coupling term can be omitted, resulting in the so-called "one-way coupling," where the structural response is first calculated and then used as input ( *f <sup>q</sup>*in Equation 69) to solve for the acoustic response. The coupled structure-acoustic system can be solved either by a direct method, or more efficiently by a modal response method which can be applied to both the structural and acoustic domains.

#### *4.3.1. Combined MCA and PROM Methods*

To demonstrate the computational effectiveness and accuracy of integrating MCA in PROM, a reanalysis was performed for a modified design where five plate thickness parameters vary; chassis and its cross links, cabin, truck bed, left door, and right door. All parameters were increased by 100% from their baseline values. The sound pressure at the driver's ear was calculated using "two-way" coupling. A structural modal frequency response was used. The acoustic response was calculated using a direct method because the size of the acoustic model is relatively small. For the structural analysis, 1050 modes were retained in the 0 to 300 Hz frequency range. The combined MCA and PROM approach was compared against the NASTRAN direct solution for a modified design where all five parameters were at their upper limits. Only one iteration was used in Equation (59) in order to get the set of once-updated mode shapes for each corner design point. The subspace basis, which includes information for all five design parameters, is therefore, represented by

$$T = \begin{bmatrix} \ \mathbf{0}\_o & T\_{0,1} & T\_{1,1} & \cdots & T\_{5,1} \end{bmatrix} \tag{70}$$

The maximum error in natural frequencies as predicted by the combined MCA and PROM method and NASTRAN, is less than 0.45% in the entire frequency range. Figure 8 indicates that the sound pressures calculated by both methods are almost identical. The computation‐ al effort for the MCA method to obtain approximate mode shapes at each corner design point is about 30 seconds. In contrast, it takes about 180 seconds for an exact eigen-solution using NASTRAN. The computational cost to construct the reduced basis (*P*in PROM and *T* in PROM+MCA) is compared in Table 2. The total cost was reduced from 1080 seconds to 330 seconds. The computational saving is more significant if the number of design parame‐ ter increases.

**Figure 8.** Comparison of sound pressure at driver's ear between combined MCA and PROM method and NASTRAN.


**Table 2.** CPU time to construct reduced basis.

of vibratory displacement and acoustic pressure. A finite-element formulation of the cou‐

0

**K p H M <sup>f</sup>** (69)

*T* = *Φ<sup>o</sup> T*0,1 *T*1,1 ⋯ *T*5,1 (70)

2 0 0

*c*

æ ö é ù - é ù é ù é ù ç ÷ ê ú - = ê ú ê ú ê ú è ø ë û ë û ë û ë û **K H M d f**

r

*S SF S S b T F F SF F q*

where the vibratory displacement *d<sup>S</sup>* and the acoustic pressure *p<sup>F</sup>* are the primary variables. The finite-element representation of the two domains consists of stiffness and mass matrix pairs *K<sup>S</sup>* , *MS* and *K<sup>F</sup>* , *M<sup>F</sup>* , respectively. The air density and wave speed are *ρ*0 and*c*0. The

The spatial coupling matrix *HSF* indicates coupling between the two domains which is usu‐ ally referred to as "two-way coupling." Due to this coupling, the combined structural-acous‐ tic system of equations is not symmetric. If the acoustic effect on the structural response is small, the coupling term can be omitted, resulting in the so-called "one-way coupling," where the structural response is first calculated and then used as input ( *f <sup>q</sup>*in Equation 69) to solve for the acoustic response. The coupled structure-acoustic system can be solved either by a direct method, or more efficiently by a modal response method which can be applied to

To demonstrate the computational effectiveness and accuracy of integrating MCA in PROM, a reanalysis was performed for a modified design where five plate thickness parameters vary; chassis and its cross links, cabin, truck bed, left door, and right door. All parameters were increased by 100% from their baseline values. The sound pressure at the driver's ear was calculated using "two-way" coupling. A structural modal frequency response was used. The acoustic response was calculated using a direct method because the size of the acoustic model is relatively small. For the structural analysis, 1050 modes were retained in the 0 to 300 Hz frequency range. The combined MCA and PROM approach was compared against the NASTRAN direct solution for a modified design where all five parameters were at their upper limits. Only one iteration was used in Equation (59) in order to get the set of once-updated mode shapes for each corner design point. The subspace basis, which includes

The maximum error in natural frequencies as predicted by the combined MCA and PROM method and NASTRAN, is less than 0.45% in the entire frequency range. Figure 8 indicates that the sound pressures calculated by both methods are almost identical. The computation‐ al effort for the MCA method to obtain approximate mode shapes at each corner design point is about 30 seconds. In contrast, it takes about 180 seconds for an exact eigen-solution

information for all five design parameters, is therefore, represented by

pled undamped problem yields the following system equations of motion [24].

2

*j*

right hand side of Equation (69) denotes the external forces.

w

0

160 Advances in Vibration Engineering and Structural Dynamics

both the structural and acoustic domains.

*4.3.1. Combined MCA and PROM Methods*

### *4.3.2. Optimization using MCA Method*

The goal here is to minimize the sound pressure at the driver's ear. A total of 41 design pa‐ rameters are used representing the thickness of all vehicle components modeled with plate elements. All thicknesses are allowed to change by 100% from their baseline values. Table 3 describes all design parameters. At the initial point of the optimization process, all parame‐ ters are at their low bound.


seconds, and the additional combined cost of Equations (62) to (64) is 373 seconds, resulting in a total cost of 404 seconds (see Table 4). To reduce this cost, the 1050 modes are divided into 21 groups and the modes in each group are obtained separately as explained in the last paragraph of Section 3.4. This reduces the cost of Equations (62) to (64) to 66 seconds for a

Vibration and Optimization Analysis of Large-Scale Structures using Reduced-Order Models and Reanalysis Methods

http://dx.doi.org/10.5772/51402

163

total cost of 97 seconds, which is about half the cost of the direct NASTRAN method.

**Figure 9.** Comparison of sound pressure at driver's ear between initial and optimal designs.

**Figure 10.** Percent increase of optimal design parameters relative to baseline design parameters.

The gradient-based optimizer in NASTRAN (SOL 200) using the optimization process of Figure 6 needed three iterations to calculate the optimal design. Figure 9 compares the

**Table 3.** Description of design parameters.

Because of the large number of design parameters, the combined MCA and PROM approach Section 4.3.1 is not computationally efficient because the size of the PROM basis is very large (see Equation 70). For this reason, we use the MCA reanalysis method and demonstrate its capability to handle a large number of parameters. It approximates the mode shapes at in‐ termediate design points using only *T*1 in Equation (59). The subspace basis at each optimi‐ zation step is thus*T* = *Φ<sup>o</sup> T*<sup>1</sup> . Because 1050 modes exist in the frequency range of 0 to 300 Hz of the initial design, the size of the MCA modal basis is 2\*1050 = 2100.


**Table 4.** CPU time of the MCA method.

The cost of solving for 1050 modes directly from NASTRAN is 180 seconds (see Table 2). In the MCA method, the cost of solving the linear system of equations in Equation (59) is 31 seconds, and the additional combined cost of Equations (62) to (64) is 373 seconds, resulting in a total cost of 404 seconds (see Table 4). To reduce this cost, the 1050 modes are divided into 21 groups and the modes in each group are obtained separately as explained in the last paragraph of Section 3.4. This reduces the cost of Equations (62) to (64) to 66 seconds for a total cost of 97 seconds, which is about half the cost of the direct NASTRAN method.

**Prm. #**

**Description (thickness of)**

162 Advances in Vibration Engineering and Structural Dynamics

**Prm. #**

**Description (thickness of)**

 Bumper 15 Radiator mtg. 29 Tire, front right Rails 16 Radiator mtg., mid. 30 Tire, rear left A-arm, low left 17 Fan cover, low 31 Tire, rear right A-arm, low right 18 Fan cover, up 32 Engine outer

 A-arm, up left 19 Cabin 33 A-arm conn., up left A-arm, up right 20 Cabin mtg. reinf. 34 A-arm conn., up right Tire rim 21 Door, left 35 A-arm conn., low left Engine Oil-box 22 Door, right 36 A-arm conn., low right

Because of the large number of design parameters, the combined MCA and PROM approach Section 4.3.1 is not computationally efficient because the size of the PROM basis is very large (see Equation 70). For this reason, we use the MCA reanalysis method and demonstrate its capability to handle a large number of parameters. It approximates the mode shapes at in‐ termediate design points using only *T*1 in Equation (59). The subspace basis at each optimi‐ zation step is thus*T* = *Φ<sup>o</sup> T*<sup>1</sup> . Because 1050 modes exist in the frequency range of 0 to 300

> Eq. (59) 31 sec 31 sec Eq. (62) 258 sec 50 sec Eq. (63) 48 sec 6 sec Eq. (64) 67 sec 10 sec Total Cost 404 sec 97 sec

The cost of solving for 1050 modes directly from NASTRAN is 180 seconds (see Table 2). In the MCA method, the cost of solving the linear system of equations in Equation (59) is 31

**k=1 k=21**

 Fan 23 Bed 37 Glass, left Hood 24 Brake, front left 38 Glass, right Fender, left 25 Brake, front right 39 Glass, rear Fender, right 26 Rail conn., rear 40 Glass, front Wheel house, left 27 Rail mount 41 Rail conn., front

Hz of the initial design, the size of the MCA modal basis is 2\*1050 = 2100.

14 Wheel house, right 28 Tire, front left

**Table 3.** Description of design parameters.

**Table 4.** CPU time of the MCA method.

**Prm. #**

**Description (thickness of)**

**Figure 9.** Comparison of sound pressure at driver's ear between initial and optimal designs.

**Figure 10.** Percent increase of optimal design parameters relative to baseline design parameters.

The gradient-based optimizer in NASTRAN (SOL 200) using the optimization process of Figure 6 needed three iterations to calculate the optimal design. Figure 9 compares the sound pressure at the driver's ear between the optimal and initial designs. Figure 10 shows the percentage increase of optimal values relative to the initial values for all 41 design pa‐ rameters. In the frequency range of 80-140Hz, the maximum sound pressure is slightly re‐ duced from 7.9E-7 to 7.2E-7 Pascal. Most parameters are minimally changed. The largest increase is 20% for the rail mount thickness (parameter #27).

The Sequential Quadratic Programming (SQP) algorithm of NASTRAN can only find a local optimum. To obtain a more significant design improvement, two additional studies were performed using a Genetic Algorithm with the MCA method. The first study used 20 initial populations and 4 generations, and the second study used 100 initial populations and 6 gen‐ erations. Figures 11and 12 show that the number of initial populations and the number of generations, affect the optimization results. While a higher number of initial populations and generations results in a slightly better result, both studies produced a much better opti‐ mum than the SQP algorithm. In the case of 100 initial populations and 6 generations, the sound pressure is reduced from 7.9E-7 Pascal to 2.0E-7 Pascal, which is equivalent to about

Vibration and Optimization Analysis of Large-Scale Structures using Reduced-Order Models and Reanalysis Methods

http://dx.doi.org/10.5772/51402

165

To verify the accuracy of the MCA approximation, the sound pressure response at the opti‐ mal design from MCA+GA with 100 initial populations and 6 generations was evaluated by both direct NASTRAN and MCA. Figure 13 shows that the MCA method is very accurate. For a similar to MCA accuracy, the original CA method needed three sets of mode shapes to form the subspace basis, requiring 90 seconds to solve the linear equations. The much larger mode basis **R** in CA increases the computational cost to calculate the triple matrix products of Equation (57). Therefore for large scale, finite-element models with a high modal density, the proposed MCA method can be more efficient compared to either a complete NASTRAN

**Figure 13.** Comparison of sound pressure at driver's ear between direct nastran and mca.

**4.4. Reanalysis in Craig-Bampton Substructuring with Interface Modes**

The FRF substructuring of Section 2.3 couples two structures using FRF information be‐ tween the coupling (interface) DOFs, and the excitation and/or response DOFs. Although this approach is very efficient, it is practical only if we have a few coupling DOFs; e.g. con‐ nection of a vehicle suspension to chassis or connection of the exhaust system to body

15 dB in sound pressure level (SPL).

analysis or the original CA method.

**Figure 11.** Comparison of sound pressure at driver's ear between baseline and optimal designs with 20 initial popula‐ tions and 4 generations.

**Figure 12.** Comparison of sound pressure at driver's ear between baseline and optimal designs with 100 initial popu‐ lations and 6 generations.

The Sequential Quadratic Programming (SQP) algorithm of NASTRAN can only find a local optimum. To obtain a more significant design improvement, two additional studies were performed using a Genetic Algorithm with the MCA method. The first study used 20 initial populations and 4 generations, and the second study used 100 initial populations and 6 gen‐ erations. Figures 11and 12 show that the number of initial populations and the number of generations, affect the optimization results. While a higher number of initial populations and generations results in a slightly better result, both studies produced a much better opti‐ mum than the SQP algorithm. In the case of 100 initial populations and 6 generations, the sound pressure is reduced from 7.9E-7 Pascal to 2.0E-7 Pascal, which is equivalent to about 15 dB in sound pressure level (SPL).

sound pressure at the driver's ear between the optimal and initial designs. Figure 10 shows the percentage increase of optimal values relative to the initial values for all 41 design pa‐ rameters. In the frequency range of 80-140Hz, the maximum sound pressure is slightly re‐ duced from 7.9E-7 to 7.2E-7 Pascal. Most parameters are minimally changed. The largest

**Figure 11.** Comparison of sound pressure at driver's ear between baseline and optimal designs with 20 initial popula‐

**Figure 12.** Comparison of sound pressure at driver's ear between baseline and optimal designs with 100 initial popu‐

increase is 20% for the rail mount thickness (parameter #27).

164 Advances in Vibration Engineering and Structural Dynamics

tions and 4 generations.

lations and 6 generations.

To verify the accuracy of the MCA approximation, the sound pressure response at the opti‐ mal design from MCA+GA with 100 initial populations and 6 generations was evaluated by both direct NASTRAN and MCA. Figure 13 shows that the MCA method is very accurate. For a similar to MCA accuracy, the original CA method needed three sets of mode shapes to form the subspace basis, requiring 90 seconds to solve the linear equations. The much larger mode basis **R** in CA increases the computational cost to calculate the triple matrix products of Equation (57). Therefore for large scale, finite-element models with a high modal density, the proposed MCA method can be more efficient compared to either a complete NASTRAN analysis or the original CA method.

**Figure 13.** Comparison of sound pressure at driver's ear between direct nastran and mca.

#### **4.4. Reanalysis in Craig-Bampton Substructuring with Interface Modes**

The FRF substructuring of Section 2.3 couples two structures using FRF information be‐ tween the coupling (interface) DOFs, and the excitation and/or response DOFs. Although this approach is very efficient, it is practical only if we have a few coupling DOFs; e.g. con‐ nection of a vehicle suspension to chassis or connection of the exhaust system to body through a few hangers. If the physical substructures have interfaces with many DOFs, a dif‐ ferent reduced-order modeling (ROM) approach must be used such as the Craig-Bampton ROM of Section 2.2.1. The Craig-Bampton ROM can be large however, if the number of re‐ tained interface DOFs is large. We address this problem by performing a secondary eigen‐ value analysis which yields the so-called *interface modes* (see Section 2.2.2). The following section describes a reanalysis methodology for physical substructuring with Craig-Bampton ROMs using interface modes. We show that its accuracy is very good and the computational savings are substantial.

*mi*

*mi*

*ki*

CCN and **k**<sup>i</sup>

CC , **m**<sup>i</sup>

tional cost related to the constraint modes.

**m***i*

**Φ ^** *i <sup>C</sup>* <sup>=</sup>**Φ***<sup>i</sup>*

*ΓΓΦ CC* + (*Φ CC*)*<sup>T</sup>*

*CCN* =(*Φ CC*)*<sup>T</sup>*

*CC* =(*<sup>Φ</sup> CC*)*<sup>T</sup>* **<sup>k</sup>***<sup>i</sup>*

**2.** In Equation (73), the number of columns of matrix (**k***<sup>i</sup>*

**^** *i*

**1.** In Equations (74) to (76), the computation involves *Φ CC* and *Φ*

*<sup>C</sup>*. Therefore, the calculation of original constraint modes *Φ<sup>i</sup>*

product in Equations (74) to (76) are now proportional to *ncc* and*ncc*

is much smaller than the corresponding cost in Equations (9), (10) and (12).

In this **CCROM** method which is based on **CBROM**, the interface modes *Φ CC* are obtained using the assembled interface partitions of the CBROM formulation. Thus, it is impossible to know *Φ CC* before hand for a new design. For this reason, Equations (73) to (76) can not be

**m***i*

**k***i*

The following observations can be made:

where the matrices **m**<sup>i</sup>

**k**i C .

**m***i*

*Φi*

for *Φ* **^** *i*

**3.** Because both *Φ CC* and *Φ*

*CC* =(*Φ CC*)*<sup>T</sup>*

*CC* =*Φ CCT <sup>T</sup>*

*CCN* =*Φ CCT <sup>T</sup>*

*CC* =*Φ CCT <sup>T</sup>*

*mi <sup>C</sup>Φ CC*

Vibration and Optimization Analysis of Large-Scale Structures using Reduced-Order Models and Reanalysis Methods

*ki <sup>C</sup>Φ CC*

The interface modes reduce the interface size producing a smaller reduced order model (ROM) compared with the traditional Craig-Bampton ROM (CBROM). However, they are calculated from the assembled interface **K** and **M** matrices. Thus, the calculation of con‐ straint modes and all matrix multiplications related to constraint modes are still necessary. The interface mode method reduces the size of ROM but it does not reduce the computa‐

If the interface modes *Φ CC* were known before hand, the calculations in Equations (6), (9),

*ΩΩ* <sup>−</sup><sup>1</sup> (**k***i*

*<sup>N</sup>* + (*Φ* **^** *i <sup>C</sup>*)*<sup>T</sup>* **<sup>m</sup>***<sup>i</sup>*

*ΓΓ<sup>Φ</sup> CC* <sup>−</sup>(*<sup>Φ</sup> CC*)*<sup>T</sup> <sup>k</sup><sup>i</sup>*

interface modes *ncc*which is usually smaller than*nc*. Therefore, the FBS cost of solving

*<sup>C</sup>* is proportional to*ncc*and it is much smaller than the FBS cost of solving for*Φ<sup>i</sup>*

(10) and (12) and Equation (72) could be performed much more efficiently as follows55

*<sup>C</sup>***Φ***CC* <sup>=</sup> <sup>−</sup>**k***<sup>i</sup>*

**m***i ΓΩΦ* **^** *i <sup>C</sup>* + (*Φ* **^** *i <sup>C</sup>*)*<sup>T</sup>* **<sup>m</sup>***<sup>i</sup>*

**m***i ΓΩΦ<sup>i</sup>* *mi CN*

CC are of much smaller size than matrices **m**<sup>i</sup>

*ΩΓΦ CC* + (*Φ*

*ΩΩΦ<sup>i</sup>*

*ΓΩΦ* **^** *i* **^** *i <sup>C</sup>*)*<sup>T</sup>* **<sup>m</sup>***<sup>i</sup>*

> **^** *i*

*<sup>C</sup>* are of size *ncc*the cost of matrix multiplication and triple-

(72)

167

CN and

C , **m**<sup>i</sup>

http://dx.doi.org/10.5772/51402

*ΩΓ***Φ***CC*) (73)

*ΩΩ*(*Φ* **^** *i*

*<sup>N</sup>* (75)

*CC* (76)

*<sup>C</sup>* and does not involve

. Therefore, the cost

*C*.

*<sup>C</sup>* is no longer needed.

*ΩΓΦ CC*) is equal to the number of

2

*<sup>C</sup>*)*<sup>T</sup>* (74)

#### *4.4.1. Craig-Bampton with Interface Modes and Reanalysis*

In the Craig-Bampton CMS method (Craig-Bampton reduced-order model or CBROM), the mass and stiffness matrices of each substructure are partitioned into interface sub-matrices, interior (omitted DOF) sub-matrices, and their coupling sub-matrices. The dynamics of a structure are then described by the normal modes of its individual components, plus a set of modes called *constraint modes* that couple the components. In CBROM, there is no size re‐ duction for constraint modes since all of them are kept in the reduced equations. If the finite element mesh is sufficiently fine, the constraint-mode DOFs will dominate the size of CBROM mass and stiffness matrices and result in a large computational cost. This issue is addressed by using *interface modes* (also called *characteristic constraint –CC- modes*). For that, a secondary eigenvalue analysis is performed using the constraint-mode partitions of the CMS mass and stiffness matrices. The CC modes are the resultant eigenvectors. Details are provided in Sections 2.2.1 and 2.2.2.

The number of constraint modes *nc* equals to the number of interface DOF. For many FE models of large structures, the number of interface DOF can be rather large. The calculation of constraint modes in Equation (6) involves a decomposition step and a FBS step. The cost of FBS is proportional to*nc*. For any matrix multiplication that involves*Φ<sup>i</sup> <sup>C</sup>*, the cost is pro‐ portional to*nc*. For any triple-product that involves *Φ<sup>i</sup> <sup>C</sup>* the cost is proportional to*nc* 2 .

The matrices from all substructures are assembled into a global CBROM with substructures coupled at interfaces by enforcing displacement compatibility. If *k<sup>i</sup> <sup>C</sup>* and *m<sup>i</sup> <sup>C</sup>* are the compo‐ nent (substructure) matrices, the global matrices *K <sup>C</sup>*and *M <sup>C</sup>*are assembled as

$$\mathbf{K}^{\mathcal{C}} = \sum \mathbf{k}\_{\iota}^{\mathcal{C}}, \quad \mathbf{M}^{\mathcal{C}} = \sum \mathbf{m}\_{\iota}^{\mathcal{C}} \tag{71a}$$

and a secondary eigenvalue analysis is performed to calculate the *interface modes Φ CC* as

$$\mathbf{[K}^{\mathcal{C}} - \lambda^{\mathcal{C}\mathcal{C}} \mathbf{M}^{\mathcal{C}}] \mathbf{[}\boldsymbol{\Phi}^{\mathcal{C}\mathcal{C}} = 0\tag{71b}$$

The matrices in Equations (9), (10) and (12) are then reduced as

Vibration and Optimization Analysis of Large-Scale Structures using Reduced-Order Models and Reanalysis Methods http://dx.doi.org/10.5772/51402 167

$$\begin{aligned} \mathfrak{m}\_{i}^{\text{CC}} &= \boldsymbol{\upPhi}^{\text{CCT}^{\text{T}}} \boldsymbol{\upm}\_{i}^{\text{C}} \boldsymbol{\upPhi}^{\text{CC}} \\ \mathfrak{m}\_{i}^{\text{CCN}} &= \boldsymbol{\upPhi}^{\text{CCT}^{\text{T}}} \boldsymbol{\upm}\_{i}^{\text{CR}} \\ \mathfrak{k}\_{i}^{\text{CC}} &= \boldsymbol{\upPhi}^{\text{CCT}^{\text{T}}} \boldsymbol{k}\_{i}^{\text{C}} \boldsymbol{\upPhi}^{\text{CC}} \end{aligned} \tag{72}$$

where the matrices **m**<sup>i</sup> CC , **m**<sup>i</sup> CCN and **k**<sup>i</sup> CC are of much smaller size than matrices **m**<sup>i</sup> C , **m**<sup>i</sup> CN and **k**i C .

The interface modes reduce the interface size producing a smaller reduced order model (ROM) compared with the traditional Craig-Bampton ROM (CBROM). However, they are calculated from the assembled interface **K** and **M** matrices. Thus, the calculation of con‐ straint modes and all matrix multiplications related to constraint modes are still necessary. The interface mode method reduces the size of ROM but it does not reduce the computa‐ tional cost related to the constraint modes.

If the interface modes *Φ CC* were known before hand, the calculations in Equations (6), (9), (10) and (12) and Equation (72) could be performed much more efficiently as follows55

$$
\stackrel{\Delta}{\mathbf{\hat{O}}}\_{i}^{\complement} = \mathbf{\hat{O}}\_{i}^{\complement} \boldsymbol{\Phi}^{\complement} = -\mathbf{k}\_{i}^{\complement\Omega^{\complement}} (\mathbf{k}\_{i}^{\amspace} \boldsymbol{\Phi}^{\complement}) \tag{73}
$$

$$\mathbf{m}\_{i}^{\circlearrowleft\mathcal{C}} = \left(\boldsymbol{\upTheta}^{\circ \text{CC}}\right)^{T} \mathbf{m}\_{i}^{\circlearrowright} \boldsymbol{\upTheta}^{\circ \text{CC}} + \left(\boldsymbol{\upTheta}^{\circ \text{CC}}\right)^{T} \mathbf{m}\_{i}^{\circlearrowright} \boldsymbol{\upTheta}\_{i}^{\circ \text{C}} + \left(\boldsymbol{\upTheta}\_{i}^{\circ \text{C}}\right)^{T} \mathbf{m}\_{i}^{\circlearrowright \mathcal{C}} \boldsymbol{\upTheta}^{\circ \text{CC}} + \left(\boldsymbol{\upTheta}\_{i}^{\circ \text{C}}\right)^{T} \mathbf{m}\_{i}^{\circ \text{C}\Omega} \left(\boldsymbol{\upTheta}\_{i}^{\circ \text{C}}\right)^{T} \tag{74}$$

$$\mathbf{m}\_{i}^{\text{CCN}} = \left(\boldsymbol{\upPhi}^{\text{CC}}\right)^{T} \mathbf{m}\_{i}^{\text{ID}} \boldsymbol{\upPhi}\_{i}^{N} + \left(\boldsymbol{\upPhi}\_{i}^{\text{C}}\right)^{T} \mathbf{m}\_{i}^{\text{
\Omega}\Omega} \boldsymbol{\upPhi}\_{i}^{N} \tag{75}$$

$$\mathbf{k}\_{i}^{\rm CC} = \begin{Bmatrix} \mathbf{\oplus}^{\rm CC} \end{Bmatrix}^{T} \mathbf{k}\_{i}^{\rm IT} \boldsymbol{\Phi}^{\rm CC} - \begin{Bmatrix} \mathbf{\oplus}^{\rm CC} \end{Bmatrix}^{T} \mathbf{k}\_{i}^{\rm ID} \boldsymbol{\hat{\Phi}}\_{i}^{\rm CC} \tag{76}$$

The following observations can be made:

through a few hangers. If the physical substructures have interfaces with many DOFs, a dif‐ ferent reduced-order modeling (ROM) approach must be used such as the Craig-Bampton ROM of Section 2.2.1. The Craig-Bampton ROM can be large however, if the number of re‐ tained interface DOFs is large. We address this problem by performing a secondary eigen‐ value analysis which yields the so-called *interface modes* (see Section 2.2.2). The following section describes a reanalysis methodology for physical substructuring with Craig-Bampton ROMs using interface modes. We show that its accuracy is very good and the computational

In the Craig-Bampton CMS method (Craig-Bampton reduced-order model or CBROM), the mass and stiffness matrices of each substructure are partitioned into interface sub-matrices, interior (omitted DOF) sub-matrices, and their coupling sub-matrices. The dynamics of a structure are then described by the normal modes of its individual components, plus a set of modes called *constraint modes* that couple the components. In CBROM, there is no size re‐ duction for constraint modes since all of them are kept in the reduced equations. If the finite element mesh is sufficiently fine, the constraint-mode DOFs will dominate the size of CBROM mass and stiffness matrices and result in a large computational cost. This issue is addressed by using *interface modes* (also called *characteristic constraint –CC- modes*). For that, a secondary eigenvalue analysis is performed using the constraint-mode partitions of the CMS mass and stiffness matrices. The CC modes are the resultant eigenvectors. Details are

The number of constraint modes *nc* equals to the number of interface DOF. For many FE models of large structures, the number of interface DOF can be rather large. The calculation of constraint modes in Equation (6) involves a decomposition step and a FBS step. The cost

The matrices from all substructures are assembled into a global CBROM with substructures

and a secondary eigenvalue analysis is performed to calculate the *interface modes Φ CC* as

*<sup>C</sup>*, the cost is pro‐

2 .

*<sup>C</sup>* are the compo‐

*<sup>C</sup>* the cost is proportional to*nc*

, *C CC C* **K kM m** = = å å *i i* (71a)

*K <sup>C</sup>* −*λ CCM <sup>C</sup> Φ CC* =0 (71b)

*<sup>C</sup>* and *m<sup>i</sup>*

of FBS is proportional to*nc*. For any matrix multiplication that involves*Φ<sup>i</sup>*

nent (substructure) matrices, the global matrices *K <sup>C</sup>*and *M <sup>C</sup>*are assembled as

coupled at interfaces by enforcing displacement compatibility. If *k<sup>i</sup>*

The matrices in Equations (9), (10) and (12) are then reduced as

portional to*nc*. For any triple-product that involves *Φ<sup>i</sup>*

savings are substantial.

provided in Sections 2.2.1 and 2.2.2.

*4.4.1. Craig-Bampton with Interface Modes and Reanalysis*

166 Advances in Vibration Engineering and Structural Dynamics


In this **CCROM** method which is based on **CBROM**, the interface modes *Φ CC* are obtained using the assembled interface partitions of the CBROM formulation. Thus, it is impossible to know *Φ CC* before hand for a new design. For this reason, Equations (73) to (76) can not be theoretically implemented to improve efficiency. For this reason, we propos*e a reanalysis ap‐ proach where the calculated interface modes Φ CC for original (baseline) design can be used as an ap‐ proximation of the new interface modes at any modified design*. In this case, Equations (73) to (76) are applied to improve the computational efficiency.

## *4.4.2. A Car Door Example*

The car door model of Figures 14 and 15 is used to demonstrate the proposed reanalysis method for substructuring with Craig-Bampton method using interface modes. It has 25,800 nodes and 25,300 elements and is divided into two substructures. The first substructure in‐ cludes the outer door shell and a bar attached to it. The second substructure includes the rest of the door. There are 293 nodes (1758 DOFs) on the interface. Therefore, the **CBROM** or **CCROM** method must calculate 1758 constraint modes according to Equation (6) for both substructures. The 1758 constraint modes are involved in matrix multiplication or tripleproducts in Equations (9), (10) and (12). Figure 16 shows the interface nodes.

**Figure 15.** Substructure 1 (outer door shell) and substructure 2 (rest of door).

Vibration and Optimization Analysis of Large-Scale Structures using Reduced-Order Models and Reanalysis Methods

http://dx.doi.org/10.5772/51402

169

**Figure 16.** Interface nodes indicated by white dots.

For the initial design using the CCROM method, 52 interface modes are calculated below 600 Hz. A modified design is created where the shell thicknesses for the outer door (substruc‐ ture 1) and inner door (substructure 2) are doubled. To provide baseline numbers, the **CCROM** method is used on the new design to solve for the system natural frequencies. The new reanalysis approach is used on the new design to calculate approximate natural frequencies which are then compared with the baseline numbers. The interface modes calculated at the original design are used as an initial guess for the interface modes of the new design.

**Figure 14.** Outside and inside views of car door model.

Vibration and Optimization Analysis of Large-Scale Structures using Reduced-Order Models and Reanalysis Methods http://dx.doi.org/10.5772/51402 169

**Figure 15.** Substructure 1 (outer door shell) and substructure 2 (rest of door).

theoretically implemented to improve efficiency. For this reason, we propos*e a reanalysis ap‐ proach where the calculated interface modes Φ CC for original (baseline) design can be used as an ap‐ proximation of the new interface modes at any modified design*. In this case, Equations (73) to (76)

The car door model of Figures 14 and 15 is used to demonstrate the proposed reanalysis method for substructuring with Craig-Bampton method using interface modes. It has 25,800 nodes and 25,300 elements and is divided into two substructures. The first substructure in‐ cludes the outer door shell and a bar attached to it. The second substructure includes the rest of the door. There are 293 nodes (1758 DOFs) on the interface. Therefore, the **CBROM** or **CCROM** method must calculate 1758 constraint modes according to Equation (6) for both substructures. The 1758 constraint modes are involved in matrix multiplication or triple-

For the initial design using the CCROM method, 52 interface modes are calculated below 600 Hz. A modified design is created where the shell thicknesses for the outer door (substruc‐ ture 1) and inner door (substructure 2) are doubled. To provide baseline numbers, the **CCROM** method is used on the new design to solve for the system natural frequencies. The new reanalysis approach is used on the new design to calculate approximate natural frequencies which are then compared with the baseline numbers. The interface modes calculated at the original design

products in Equations (9), (10) and (12). Figure 16 shows the interface nodes.

are used as an initial guess for the interface modes of the new design.

**Figure 14.** Outside and inside views of car door model.

are applied to improve the computational efficiency.

168 Advances in Vibration Engineering and Structural Dynamics

*4.4.2. A Car Door Example*

**Figure 16.** Interface nodes indicated by white dots.

In the new approach to reduce the cost related to constraint modes, the total remaining cost is dominated by the cost of calculating the normal modes for each substructure. For exam‐ ple, the calculation of the normal modes for Substructure 2 took 110 seconds out of a total of 139 seconds (see Table 5). It should be noted that the normal modes cost can be further re‐ duced by applying another reanalysis method such as CDH/VAO, CA or MCA to approxi‐ mate the normal modes. Therefore, the overall cost of substructuring based on Craig-Bampton with interface modes, can be drastically reduced by using the proposed reanalysis to approximate the constraint modes and a CDH/VAO or MCA reanalysis to approximate

Vibration and Optimization Analysis of Large-Scale Structures using Reduced-Order Models and Reanalysis Methods

http://dx.doi.org/10.5772/51402

171

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

the normal modes at a new design.

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

**5. Optimization of a Vehicle Model**

and 1.1 million elements. Figure 18 shows all modeling details.

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

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


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

In the new approach to reduce the cost related to constraint modes, the total remaining cost is dominated by the cost of calculating the normal modes for each substructure. For exam‐ ple, the calculation of the normal modes for Substructure 2 took 110 seconds out of a total of 139 seconds (see Table 5). It should be noted that the normal modes cost can be further re‐ duced by applying another reanalysis method such as CDH/VAO, CA or MCA to approxi‐ mate the normal modes. Therefore, the overall cost of substructuring based on Craig-Bampton with interface modes, can be drastically reduced by using the proposed reanalysis to approximate the constraint modes and a CDH/VAO or MCA reanalysis to approximate the normal modes at a new design.
