An improved reanalysis method using parametric reduced order modeling for linear dynamic systems.
Finite element analysis is a standard tool for deterministic or probabilistic design optimization of dynamic systems. The optimization process requires repeated eigenvalue analyses which can be computationally expensive. Several reanalysis techniques have been proposed to reduce the computational cost including Parametric Reduced Order Modeling (PROM), Combined Approximations (CA), and the Modified Combined Approximations (MCA) method. Although the cost of reanalysis is substantially reduced, it can still be high for models with a large number of degrees of freedom and a large number of design variables. Reanalysis methods use a basis composed of eigenvectors from both the baseline and the modified designs which are in general linearly dependent. To eliminate the linear dependency and improve accuracy, Gram Schmidt orthonormalization is employed which is costly itself. In this paper, we propose a method to reduce the orthonormalization cost and improve the computational efficiency of the PROM reanalysis method. Our method eliminates non-important design variables and/or reduces the basis size by eliminating redundant modes. A vibratory analysis of an automotive door demonstrates the efficiency and accuracy of the proposed method.
CITATION: Haider, S. and Mourelatos, Z., "An Improved Reanalysis Method Using Parametric Reduced Order Modeling for Linear Dynamic Systems," SAE Int. J. Passeng. Cars - Mech. Syst. 9(1):2016.
Finite element analysis (FEA) is often the main computational tool for Noise, Vibration and Harshness (NVH) studies. Although FEA can handle large-scale models, automotive companies demand even larger detailed models with millions of degrees of freedom (DOFs) to study vibro-acoustic problems. As a result, the computational cost can be prohibitive even for high-end workstations with the most advanced software.
For linear vibration analysis, a modal reduction is commonly used to solve for the system response. An eigenanalysis is performed using the system stiffness and mass matrices and a smaller in size modal model is formed which is solved more efficiently. The computational cost can also be reduced using substructuring (superelement analysis). Modal reduction is applied to each substructure to obtain the component modes and the system level response is obtained using Component Mode Synthesis (CMS).
When design changes are involved (e.g., optimization), the FEA analysis must be repeated many times and the computational cost can become prohibitive. Reanalysis methods are intended to efficiently analyze structures that are repeatedly modified by estimating the response of the modified structure without solving the complete set of modified analysis equations. They use a basis of eigenvectors from the baseline design, augmented sometimes by eigenvectors from other design points as in PROM , to estimate the eigenvalues and eigenvectors of the modified design without performing a new eigenvalue analysis. The eigenvectors in the basis are often linearly dependent. To avoid numerical errors from this dependency, a Gram Schmidt orthonormalization must be performed. In general, this is a very costly operation especially when the basis is large (many eigenvectors are used) and/or the model size is large (many DOFs).
The eigenvalue problem for an undamped system with stiffness [K.sub.nxn] and mass [M.sub.nxn] is given by
[K.sub.nxn] [[PHI].sub.nxm] = [M.sub.nxn] [[PHI].sub.nxm] [[LAMBDA].sub.mxm] (1)
where n represents the number of degrees of freedom (DOF) and m represents the number of kept modes. If a perturbation is applied on the existing structure, the modified system matrices are
[K.sub.p] = [K.sub.0] + [DELTA]K, [M.sub.p] = [M.sub.0] + [DELTA]M (2)
where [K.sub.0], [M.sub.0] and [K.sub.p], [M.sub.p] are the stiffness and mass matrices of the original (baseline) and modified systems, respectively.
In reanalysis, a modal model is formed using the following reduced matrices
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where [[PHI].sub.0] are the eigenvectors of the original (baseline) design and the following eigenvalue problem is then solved
[K.sub.r][THETA] = [LAMBDA][M.sub.r][THETA]. (4)
The reduced eigenvector [THETA] is then projected back to the physical domain to obtain the eigenvectors [[PHI].sub.p] of the modified design as
[[PHI].sub.p] = [[PHI].sub.0] [THETA]. (5)
In this paper, we present an approach to minimize the computational effort for the eigenvector orthonormalization in PROM by reducing the size of the basis. This is achieved first by eliminating the eigenvectors of non-important design variables (corner points) and second by neglecting redundant modes. In Section 2, we present an overview of reanalysis methods, including PROM. Section 3 provides details on our proposed methodology and Section 4 uses an automotive door example to demonstrate the efficiency and accuracy of the proposed approach. Finally, Section 5 summarizes and concludes.
2. REVIEW OF EXISTING REANALYSIS METHODS
In this section, we briefly review the PROM, CA, and MCA reanalysis methods.
Parametric Reduced-Order Modeling (PROM)
The PROM method approximates the mode shapes (eigenvectors) of a new design in the subspace spanned by the dominant mode shapes of some representative designs. The latter are selected such that the formed basis captures the dynamic characteristics of the modified design due to changes in each dimension of the parameter space (Figure 1). Balmes [2, 3] suggested that these representative designs should correspond to the mid-range of design parameters in the parameter space.
For a structure with m design variables, Zhang  suggested that the representative designs include a baseline design for which all parameters are at their lower limits plus m designs obtained by perturbing the design variables from their lower limits to their upper limits, one at a time. The points representing these designs in the space of the design parameters will be called corner points (Figure 1).
The mode shapes [PHI] of a new design are approximated in the space of the mode shapes of the corner points as
[PHI] [approximately equal to] [[??].sub.p] = P[THETA] (6)
where the modal matrix P contains the basis vectors according to Equation (7)
P=[[[PHI].sub.0] [[PHI].sub.1] ... [[PHI].sub.m]] (7)
and [THETA] represents the participation factors of these vectors. The columns of P are the dominant mode shapes of the above (m + 1) designs. Figure 1 shows a design space for three design parameters. Point [P.sub.0] represents the baseline design with all design parameters at their minimum values, and point [P.sub.i], i = 1,2,3 represents a modified design with all parameters at their minimum values except parameter i which is at its maximum (perturbed) value.
In Equation (7), [[PHI].sub.0] is the modal matrix composed of the dominant mode shapes of the baseline design, and [[PHI].sub.i] is the modal matrix of the ith corner point. The mode shapes of the new design satisfy the following eigenvalue problem,
K[[??].sub.p] = [LAMBDA]M[[??].sub.p] [left right arrow] KP[THETA] = [LAMBDA]MP[THETA] (8)
where K and M are the stiffness and mass matrices of the modified design and [LAMBDA] is a diagonal matrix of the first m eigenvalues.
A reduced eigenvalue problem is obtained by pre-multiplying both sides of Equation (8) by [P.sup.T] to obtain
[K.sub.R][THETA] = [LAMBDA][M.sub.R][THETA] (9)
where the reduced stiffness and mass matrices are
[K.sub.R] = [P.sup.T]KP and [M.sub.R] = [P.sup.T]MP. (10)
The matrix [THETA] in Equation (9) contains the eigenvectors of the reduced stiffness and mass matrices [K.sub.R] and [M.sub.R].
For m design variables, (m + 1) eigenvalue problems must be solved in order to form the basis P of Equation (7). Therefore, both the cost of obtaining the modal matrices [[PHI].sub.i] and the size of matrix P increase linearly with m. The PROM approach uses the following algorithm to compute the mode shapes of a new design:
1. Calculate the mode shapes of the baseline design and the designs corresponding to the m corner points in the design space, and form subspace basis P.
2. Calculate the reduced stiffness and mass matrices [K.sub.R] and [M.sub.R] using Equation (10).
3. Solve the eigen-problem of Equation (9) for matrix [THETA].
4. Reconstruct the approximated eigenvectors in [??]p using Equation (6).
In the above procedure, step 1 is performed only once. A reanalysis requires only steps 2 to 4. For a small number of mode shapes and a small number of design parameters, the cost of steps 2 to 4 is much smaller compared to the cost of a full analysis.
The computational cost of PROM consists of a) the cost of performing (m + 1) full eigen-analyses to form the subspace basis P in Equation (7), and b) the cost of reanalysis of each new design in steps 2 to 4. The former is considered the fixed cost of PROM because it does not depend on the number of reanalyses and the latter represents the variable cost of PROM since it is proportional to the number of reanalyses. The fixed cost is not attributed to the calculation of the response of a particular design. Simply, it is required to obtain the needed information to apply PROM. The variable cost (cost of reanalysis of a new design in part b) is small compared to the fixed cost.
The fixed cost of PROM is proportional to the number of design variables m since the basis P consists of the dominant eigenvectors [[PHI].sub.0] of the baseline design, and the dominant eigenvectors [[PHI].sub.i], i = 1, ..., m of the m corner design points as shown in Equation (7). As the size of P increases, so does the fixed cost because more eigenvalue problems and mode shapes must be calculated. The PROM method results in significant cost savings when applied to problems that involve few design variables (less than 20) and require many analyses (e.g. Monte Carlo simulation or gradient-free optimization using genetic algorithms).
The efficiency of the PROM reanalysis method can be further improved by integrating it with the MCA method .
Combined Approximations (CA)
In the CA method [6, 7, 8, 9, 10, 11, 12, 13, 14, 15], the eigenvectors at a new design are approximated using a linear combination of basis vectors. A subspace basis is formed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
where s is the number of basis vectors normally in the range of 3-6 depending on accuracy and efficiency [7, 10].
For a modified design, the stiffness and mass matrices are given by
[K.sub.p]=[K.sub.0]+ [LAMBDA]K, [M.sub.p],=[M.sub.0]+[LAMBDA]M (12)
where [K.sub.0] and [M.sub.0] are the stiffness and mass matrices of the baseline (original) design and [DELTA]K and [DELTA]M represent large perturbations. The basis vectors in Equation (11) are provided by Equations (13) and (14) as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
where [[PHI].sub.0] are the eigenvectors of the baseline design. For the perturbed (new) design, a reduced model is obtained with the following stiffness and mass matrices
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
and an eigenvalue problem is solved using the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] matrices of Equation (15) to calculate the eigenvectors [THETA] of the reduced model. These eigenvectors are then projected back to the physical domain to obtain the following approximate eigenvectors [[PHI].sub.p] of the new design as
[[PHI].sub.p] = [R.sub.nxs] [THETA]. (16)
Because the basis vectors are linearly dependent, a Gram Schmidt orthonormalization is performed to avoid numerical errors. The CA method is efficient for problems where the number of desired modes is low. In such a case, the computational effort of stiffness matrix decomposition (Equations 13 and 14) dominates.
The CA method has the following three main advantages:
1. It only requires a single matrix decomposition of stiffness matrix [K.sub.0] to calculate the subspace basis R,
2. It is accurate because the basis is updated for every new design, and
3. The eigenvectors of a new design are efficiently approximated using Equation (16) where the eigenvectors [THETA] correspond to a much smaller eigen-problem.
It should be noted that the CA method is more suitable than PROM, if the number of reanalyses is less than the number of design parameters. However, it is efficient only for problems with a small number of retained modes.
Modified Combined Approximations (MCA)
A modified combined approximations (MCA) method was proposed in [5, 16] where the columns of the subspace basis are obtained from by the following recursive equations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
where [K.sub.p] and [M.sub.p] are the stiffness and mass matrices of the modified design. The subspace basis is defined as
T = [[PHI].sub.0] [T.sub.1]] ... [[T.sub.s]] (19)
where s is usually equal to 1 or 2. A Gram Schmidt orthonormalization is again performed to avoid numerical errors from linear dependency. A reduced model is defined using the following reduced stiffness and mass matrices
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
and the eigenvalue problem [K.sup.R][THETA]=[[lambda].sub.0][M.sup.R][THETA] is solved to obtain the eigenvectors [THETA] which are then used to approximate the eigenvectors of the modified design as
[[PHI].sub.p] = T[THETA]. (21)
Although the MCA method has a slightly higher matrix decomposition cost as the matrix [K.sub.p] must be decomposed for each new design, the overall computational effort of MCA is lower than CA because the forward-backwards cost in solving Equations (17) and (18) is much lower than that of CA. For this reason, MCA is computationally more efficient than CA for large size models with many modes.
3. PROPOSED APPROACH
As we have mentioned, reanalysis methods use a basis of eigenvectors which must be orthonormalized to avoid linear dependencies among approximated eigenvectors. They also use a triple matrix product as in Equation (20) to create a reduced in size modal model. Both the orthonormalization and the triple matrix product operations are computationally demanding. Our proposed approach reduces the computational effort of both the orthonormalization and the triple matrix product operations in PROM by reducing the size of the basis (Equation 7). The PROM method is the only reanalysis method where the size of the basis is proportional to the number of design parameters. The CA and MCA methods have relatively smaller bases. However as all reanalysis methods (CA, MCA, and PROM) use a basis composed of approximate eigenvectors, the approach proposed in the ensuing sections is equally applicable to all. We should note that the proposed approach reduces only the PROM fixed cost.
Orthonormalization is critical in obtaining accurate results because the approximate eigenvectors can be linearly dependent. Considering that the computational effort to orthonormalize the basis increases with its size, we can simply reduce the effort by reducing the size of the basis. We propose to do so by eliminating non-important design parameters and some (or all) modes corresponding to certain design parameters. The following two subsections provide details.
Elimination of Design Parameters
To eliminate a design parameter (DP), the normal modes of the original model (DP's at their minimum values) are compared with the normal modes of models with each DP at its maximum value ("corner points" of parameter space). Computing the sum of the square of the differences and sorting in descending order, the design parameters with low sums can be eliminated reducing therefore, the size of the basis. Figure 2 outlines the process. The following steps are executed:
1. Perform normal modes analysis for the baseline model (all DP's at their minimum values).
2. Compute the normal modes of models with one DP at its maximum value and all other DP's at their minimum values.
3. Compute the sum of square of difference between the natural frequencies of the base model (step 1) and the models of step 2.
4. Sort the sums in ascending order.
5. Eliminate the DP's for which the sum of square of difference of natural frequencies is lower than a cutoff value.
6. Create a basis using the eigenvectors of the models with each of the retained DP's.
Steps 1, 2 and 6 are required by PROM. Only steps 3 through 5 are additional. Considering that the computational effort for steps 3 through 5 is minimal, the proposed approach can reduce the overall computational effort considerably.
Elimination of Modes
In order to reduce the size of the basis by eliminating certain modes, we propose using the Modal Assurance Criterion (MAC) between the baseline modes and the modes of the corner points. By setting a MAC cutoff value, all eigenvectors having a MAC value lower than the cutoff value can be eliminated. The final basis can be therefore, reduced considerably, increasing the computational efficiency. The proposed process is outlined below:
1. Calculate the eigenvalues and eigenvectors of the baseline model.
2. Calculate the eigenvalues and eigenvectors of the model with the [j.sup.th] DP at its maximum value and all others at their minimum values (j-DP model).
3. Calculate the MAC value of each mode of the baseline model with all modes of the j-DP model if the difference between eigenvalues is less than 3 Hz.
4. If the MAC value is greater than a cutoff value, eliminate the corresponding eigenvector from the basis.
5. Update the basis to reflect the mode elimination from step 4 for the j-DP model.
6. Repeat steps 2 to 5.
7. Update the basis.
8. Orthonormalize the basis to eliminate linear dependencies.
Because the MAC computation is costly, step 3 reduces the computation time by calculating the MAC for only a limited number of eigenvectors of the j-DP model. Figure 3 explains all steps of the process.
4. AUTOMOTIVE DOOR EXAMPLE USING PROM
The automotive door example of Figure 4 is used to demonstrate the realized computational savings from the proposed method. The model has 146,589 DOFs and is constrained at the door handle and the hinges. Twenty design parameters are used out of which 18 are thickness and 2 are lumped masses.
Table 1 lists the design parameters and their minimum (baseline) and maximum values. Twenty design variables are selected, out of which, 18 are gage thickness and 2 are masses. The maximum values in Table 1 are much larger than the baseline values indicating large perturbations.
The second column of Table 2 provides the number of columns (eigenvectors) in the PROM basis corresponding to the baseline design (2nd row) and the 20 "corner" points (3rd to 22nd rows). The third column shows the sum of error squared for each run. The overall PROM basis comprises 3568 columns (last row) obtained from 21 runs in the frequency range of 0 to 200 Hz. Its orthonormalization required 843 minutes (last row of 2nd column of Table 2) using a Pentium Dual core 2.8GHz with 2GB RAM. Table 3 tabulates the sorted design parameters according to the sum of error squared. Based on our previous discussion, we can eliminate the DPs with low sum of error squared. Three different bases were considered. The first basis does not include the eigenvectors for design parameters 4, 20, and 13 (last 3 DPs in Table 3). The second basis does not include the eigenvectors for design parameters 4, 20, 13, 12, 8, and 7 (last 6 DPs in Table 3) and the third basis does not include the design parameters 4, 20, 13, 12, 8, 7, 16, 2, 18, and 17 (last 10 DPs in Table 3).
As shown in Table 4, the basis size for the three cases is 3040, 2513, and 1810 respectively. The corresponding computational time for othronormalization compared to the baseline 843 minutes is 572 (32% improvement), 327 (61% improvement), and 134 (84% improvement) minutes, respectively. This is a sizable reduction in time from elimination of the non-important design parameters.
To demonstrate the accuracy after the eigenvector elimination of the three basis sets of Table 4, a unit excitation is applied on the outer (Point 1) and inner (Point 2) panels of the door (Figure 4) and the magnitude of the velocity frequency response is calculated using PROM. Figures 5 and 6 show the calculated point FRFs at the two locations on the door. We compare the results when the eigenvectors corresponding to the corner points of Set #1, Set #2 or Set #3 parameters (Table 4) are removed from the PROM basis with the case where no eigenvectors are removed (full run). For all cases, the design parameters are at their corner points (maximum values). We observe that the accuracy of all three cases is good.
The same example is used to demonstrate the effectiveness of eliminating eigenvectors using the MAC criterion. Table 5 shows the number of eigenvectors for each of the PROM corner points, after some eigenvectors are eliminated using a MAC cutoff value (2nd, 3rd and 4thcolumns). It also shows (last row) the required time for mode elimination and orthonormalization of the basis. Using MAC values of 0.85 and 0.75 for example, the time to eliminate the eigenvectors and orthonormalize the basis was 272 minutes (67% improvement) and 160 minutes (81% improvement), respectively. For comparison, if we do not eliminate any modes, the time for orthonormalization is 843 minutes.
Figures 7 and 8 show the point FRFs for the two points on the door (inner and outer panels) after the elimination of eigenvalues using a MAC value of 0.75 and 0.85. The FRFs are compared with the case where no eigenvectors are eliminated (full run). Comparison of Figures 7 and 8 with Figures 5 and 6 indicate a minimal loss of accuracy from the mode elimination using the MAC criterion.
5. SUMMARY AND CONCLUSIONS
An efficient approach to improve the fixed cost of reanalysis using PROM was presented. The developed procedure involves reducing the size of basis for estimating the eigenvectors of a new design using PROM 1) by eliminating non-important design parameters and 2) by eliminating modes of certain design parameters using the MAC criterion. The reduced size basis improves the efficiency of the orthonormalization process considerably. An automotive door example was used to demonstrate the accuracy and efficiency advantages of the proposed approach. Future research will focus on applying the proposed approach to the CA and MCA reanalysis methods.
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Syed F. Haider and Zissimos Mourelatos
Zissimos P. Mourelatos
Professor, Mechanical Engineering Department Oakland University, Rochester, MI
Syed F. Haider
PhD Candidate, Mechanical Engineering Department Oakland University, Rochester, MI
Table 1. Design parameters for automotive door # Design Parameter Thickness (mm) 1 Inner Door panel RHS 2 Door impact bar 3 Inner Door panel LHS 4 Upper door hinge reinforcement 5 Door surround rear 6 Outer panel 7 Door latch reinforcement 8 Door surround rear 9 Door belt inner 10 Glass channel mounting bracket 11 Door surround rear 12 Inner/Outer panel skirt 13 Inner/Outer panel skirt 14 Door belt outer 15 Door surround glass 16 Impact bar bracket 17 Door Inner belt skirt 18 Glass runner Mass (Tons) 19 Side view mirror mass 20 Door trim mass # Min. Max. 1 1 3 2 1 2.75 3 0.5 1.75 4 1 3 5 0.25 1.25 6 0.25 1.25 7 1 2 8 1.25 3.25 9 0.75 1.75 10 1.25 2.25 11 0.25 1.25 12 1 3 13 2 4 14 0.25 1.25 15 0.25 1.25 16 1 3 17 1 3 18 0.25 1 19 0.001 0.002 20 0.00035 0.0005 Table 2. Size of PROM basis and sum of errors squared No. of basis columns Sum of error DP for PROM baseline squared and "corner" points Base 176 Model 1 173 2965.82 2 176 78.83 3 164 18147.88 4 176 0.56 5 175 115.34 6 76 370152.8 7 175 49.56 8 176 49.47 9 174 867.31 10 176 151.00 11 175 121.24 12 176 44.23 13 176 10.97 14 170 3003.77 15 175 330.78 16 176 73.11 17 176 105.51 18 175 111.67 19 176 115.21 20 176 5.53 Total # of basis 3568 columns Table 3. Sorted sum of error squared DP Sorted Sum of Error Squared 6 370152.80 3 18147.88 14 3003.77 1 2965.82 9 867.31 15 330.78 10 151.00 11 121.24 5 115.34 19 115.21 18 111.67 17 105.51 2 78.83 16 73.11 7 49.56 8 49.47 12 44.23 13 10.97 20 5.53 4 0.56 Table 4. Basis vector selection BASIS Eliminated Columns in Time (min) Set# Design PROM for Ortho- Parameters Basis normalizati on 1 4,20,13 3040 572 2 4,20,13,12,8,7 2513 327 3 4,20,13,12, 1810 134 8,7,16,2,18,17 Table 5. Mode elimination for basis reduction using MAC Size Reduction No. of basis columns for MAC MAC DP baseline and 0.75 0.85 "corner" points Base Model 176 176 176 1 173 131 155 2 176 46 68 3 164 143 154 4 176 0 3 5 175 48 79 6 76 67 72 7 175 26 43 8 176 20 33 9 174 110 130 10 176 32 37 11 175 49 73 12 176 34 49 13 176 5 11 14 170 109 139 15 175 59 91 16 176 26 54 17 176 33 59 18 175 18 39 19 176 4 11 20 176 2 2 Total Basis 3568 1138 1478 columns % Reduction 68.11 58.58 Time (min) for Ortho- normalization 843 160 272 and mode elimination
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|Author:||Haider, Syed F.; Mourelatos, Zissimos|
|Publication:||SAE International Journal of Passenger Cars - Mechanical Systems|
|Date:||Apr 1, 2016|
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