have invested in supercomputer systems, which have cost up to tens of millions of dollars each. Over the past year or so, however, the future of vecto

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1 MEETING THE NVH COMPUTATIONAL CHALLENGE: AUTOMATED MULTI-LEVEL SUBSTRUCTURING J. K. Bennighof, M. F. Kaplan, y M. B. Muller, y and M. Kim y Department of Aerospace Engineering & Engineering Mechanics The University of Texas at Austin Austin, Texas ABSTRACT Noise, Vibration and Harshness (NVH) analysis has presented a formidable computational challenge in the last several years. Accurate prediction of automobile frequency response at higher frequencies requires FE models which have millions of DOF. For obtaining frequency response over a broad frequency range, only the modal frequency response approach has been practical, and only vector supercomputers have been capable of solving for natural frequencies and modes in an acceptable amount of time. However, these machines are very expensive, and their future availability is uncertain. In the automated multi-level substructuring (AMLS) method, a large FE model is automatically divided into many substructures, and the model is transformed so that response is represented in terms of substructure eigenvectors. The numerical example demonstrates that the frequency response of a 2.9 million DOF FE car body model can be obtained more quickly using a single workstation processor with AMLS than it can be obtained using a supercomputer and the modal frequency response approach. NOMENCLATURE! : radian excitation frequency M; B; K;?: mass, viscous damping, stiness, structural damping matrices u; F : vectors of FE nodal displacements, forces : global structural damping parameter L; S : substructure \local," \shared" DOFs, partitions : matrix of substructure xed-interface eigenvectors : matrix of substructure \constraint modes" p : vector of substructure modal coordinates Associate Professor (bennighof@mail.utexas.edu). y Graduate Research Assistant. 1.INTRODUCTION An important objective of NVH (noise, vibration and harshness) frequency response FE analysis is prediction of noise levels as high as possible in the audible frequency range. This enables passenger comfort and perceived quality to be improved based on numerical simulation, rather than on costly and timeconsuming prototyping and testing. Until very recently, largescale NVH analysis has only been practical on vector supercomputers. The frequency range that could be addressed has been limited by the expense of processing on these machines, and by the amount of computer time required. In order to obtain acceptable accuracy at higher frequencies, FE models with over one million DOF are constructed. Because frequency response is desired at many frequencies, direct or iterative solutions in terms of all of the FE DOF are not feasible, so the modal frequency response approach is used instead. In the modal frequency response approach, a partial eigensolution is computed for the entire FE model, to obtain eigenpairs up to a cuto frequency based on the highest desired response frequency. Then the frequency response problem is projected onto the subspace of computed global eigenvectors. For the eigensolution, the block Lanczos eigensolver is used in conjunction with a sparse direct (out-of-core) solver. Unfortunately, attempts to parallelize the Lanczos algorithm for problems of this size have met with very limited success. Another obstacle is that for the Lanczos algorithm, the number of oating point operations per memory reference is relatively low. For this reason, a computer must have a high memory bandwidth to keep processors supplied with data. As a result, vector supercomputers have proven to be much faster for this algorithm than microprocessor-based systems. In the last several years, nearly all of the major automakers 1

2 have invested in supercomputer systems, which have cost up to tens of millions of dollars each. Over the past year or so, however, the future of vector supercomputers has become questionable as multiprocessor machines, with more favorable price/performance ratios on most applications, have become available, and commercial software has emerged which can effectively exploit capabilities of these machines. As the number of applications on which supercomputers oer a substantial performance advantage has decreased, the future economic viability of these machines has become less certain. In short, NVH analysis by conventional methods requires very expensive supercomputer hardware that may not be available for much longer. The size of FE models that can be handled and the frequency range that can be addressed are limited by the computational expense associated with frequency response analysis. Over the last few years, a new method known as automated multi-level substructuring (AMLS) has been developed for frequency response analysis of large FE models.[1,2] In AMLS, the FE model is automatically subdivided into thousands of substructures in a tree topology. Frequency response is represented in terms of substructure eigenvectors rather than global modes of vibration. Probably the greatest advantage of AMLS is that it enables frequency response analysis to be done on microprocessor-based systems, including workstations, rather than supercomputers. Because AMLS requires much less memory bandwidth and fewer oating point operations than the conventional modal frequency response approach, large jobs require less time on a single workstation processor using AMLS than they require on a supercomputer using NASTRAN software, as is demonstrated in the numerical example for this paper. AMLS has much lower disk space requirements than the modal frequency response approach, because in the conventional approach, thousands of eigenvectors and iteration vectors, expressed in all of the FE DOF, must be written to disk. In contrast, in AMLS it is not necessary to produce eigenvectors in all FE DOF. As a result, large jobs can be processed with tens of gigabytes of workstation disk space, rather than the much larger amounts typical on supercomputers. AMLS is also inherently more parallelizable than the Lanczos algorithm. Dierent substructures can be processed simultaneously and independently. Also, as transformation of the FE model to the substructure eigenvector representation progresses toward the root of the substructure tree, there is more opportunity for parallelizing the computation associated with each individual substructure. Evidently, with the capabilities that AMLS brings, there is potential for transforming the NVH enterprise from its present state, in which FE models are limited in size to about 2 million DOF and must be processed in batch on supercomputer systems. Jobs require so much CPU time and disk space that only a relatively small number of analyses can be performed to assess and improve NVH performance. The limitation on model size can be increased dramatically, improving accuracy and frequency range that can be addressed using models created with a high level of automation. The use of inexpensive microprocessor-based systems with multiple processors will enable rapid job turnaround and much more opportunity to optimize NVH performance. In the next section, the procedure for transforming FE models to the substructure mode representation is described. The following section gives an overview of procedures for approximating a structure's frequency response and global eigensolution after its model has been transformed. For a numerical example, AMLS is used to solve for the frequency response of a model having nearly 3 million DOF, and the results are compared with those obtained using commercial NASTRAN software on a supercomputer. Conclusions are summarized in the nal section. 2. MULTI-LEVEL SUBSTRUCTURING Frequency response of a FE model of a structure is typically governed by a system of equations of the form??! 2 M + i!b + (1 + i)k + i? u = F : (1) Here, damping is modeled using several possible approaches, including viscous damping (matrix B), structural damping that is globally uniform (scalar ), or structural damping which deviates locally from the global structural damping level (matrix?), typically for elements whose material properties dier from those of the predominant structural material. Damping can also be modeled using constant or frequency-dependent modal damping factors in modal frequency response computation. The FE model is transformed so that response is represented in terms of substructure modes. The rst step is to partition the model automatically into substructures on a number of levels, based on the sparsity structure of the system matrices. Substructures on the lowest level consist of a small number of nite elements. These \child" substructures are assembled together to form \parents," which are assembled together to form \grandparents," and so on until the model of the complete structure has been assembled. This results in a \tree" topology for the substructures, whose highest levels are shown in Figure 1 for the case in which each substructure has two \children". For large FE models, substructures typically number at least in the thousands. Within a substructure on the lowest level, displacement degrees of freedom are partitioned into two sets. One set consists of both \interface" and \forced" degrees of freedom, i.e., those that are shared with adjacent substructures at interfaces, and those that correspond to nonzeros in the force vector. These are called \shared" degrees of freedom, since the substructure shares these directly with its environment, in a general sense. 2

3 where I represents an identity submatrix. Damping matrices B and/or? are transformed similarly. Figure 1. Substructure tree. The remaining degrees of freedom are called \local" degrees of freedom. They are not excited directly, but only through coupling with the \shared" degrees of freedom. The substructure displacement vector u is partitioned into u S and u L. In AMLS, local degrees of freedom are represented with a quasistatic dependence on shared degrees of freedom, with deviation from quasistatic dependence represented using substructure \xed interface mode" eigenvectors. This combination of quasistatic dependence and modal response was used by Hurty [3] in a form which explicitly preserved substructure rigid body modes. Craig and Bampton [4] observed that substructure rigid body modes do not need to be preserved explicitly, and this led to a more streamlined formulation. Substructure response is represented in Craig-Bampton form as u = u L L L p p = = [ ] u S 0 I u S u S ; (2) where L satises the algebraic eigenvalue problem K LL L = M LL L L. Here, substructure stiness and mass matrices K and M are partitioned as u is partitioned, and L is a diagonal matrix of eigenvalues. The submatrix L is given by L =?K?1 LL K LS, and p is a vector of substructure modal coordinates. As a result of this transformation from the substructure's FE model to the quasistatic/modal representation, the substructure stiness matrix becomes, assuming that eigenvectors are massnormalized, ~K = [ ] T K LL K LS K SL K SS [ ] = L 0 0 ~ K ; (3) where o-diagonal submatrices are null due to the denition of. The substructure mass matrix is transformed to ~M = [ ] T M LL M LS I M ~ [ ] = ; M SL M SS ~M ~M (4) Once lowest-level \child" substructures have been transformed, they are assembled together to form \parent" substructures on the next level. Since local child degrees of freedom are represented implicitly in terms of shared degrees of freedom by default, parent degrees of freedom are considered to consist only of child shared degrees of freedom. From these, local and shared degrees of freedom for parents are identied, and parent models are transformed as the child substructure models were. Parent \modes" are not intended to be accurate approximations of physical modes of vibration. Instead, they constitute an ecient reduced subspace with useful orthogonality properties. Assembly to form higher-level substructures, and transformation to the quasistatic/modal representation, continues until a model for the entire structure has been assembled. If all \forced" degrees of freedom are included as \shared" degrees of freedom at all levels, all degrees of freedom that have been \local" for lower-level substructures implicitly depend quasistatically on forced and other shared degrees of freedom in the last assembled model. As a result, static response of the structure is represented exactly without using any modes of lower-level substructures. Once a model for the entire structure has been assembled, its degrees of freedom that are not forced are treated in the same manner as local degrees of freedom for lower-level substructures, and forced degrees of freedom are treated as the only shared degrees of freedom. A partial eigensolution is found and the model is transformed as described above. After this transformation, the only remaining explicit degrees of freedom in the model are forced degrees of freedom. One last eigensolution is found, in which only the forced degrees of freedom appear. Response of the structure can then be expressed entirely in terms of modes of substructures on all levels. The cost of performing this transformation consists of the costs of obtaining the and matrices, and of transforming M, K, and possibly B and/or?. A comparison with the cost of sparse direct methods can be made easily because of the similarity between multi-level substructuring and the multifrontal sparse solution approach [5]. In the latter approach, as described in [6], degrees of freedom internal to individual elements are rst eliminated by static condensation. Then pairs of elements are assembled together and degrees of freedom that become internal to the pairs are similarly eliminated. Pairs of superelements are assembled together on higher and higher levels, with elimination of internal degrees of freedom at each level, until the entire structure FE model has been assembled. If substructures on multiple levels correspond to the superelements generated during the multifrontal solution process, generation of substructure matrices and transformation of K requires operations very similar to those required for sparse factorization of K. The main dierence is that, in AMLS, at every 3

4 elimination of degrees of freedom by static condensation, a partial eigensolution is obtained. The substructure modes that are computed can then be used to augment the sparse factorization so that it can be used to compute response at nonzero frequency. The partial eigensolutions are economical because substructure eigenvalue problems are very small, and their eigenvalues are typically well separated, resulting in rapid convergence. The fact that substructure stiness matrices have already been factored can be exploited. In implementation, a cuto frequency for substructure natural frequencies is selected based on the response problems of interest. Typically many substructures do not have any natural frequencies below the cuto frequency. The number of eigenvectors found determines the cost of obtaining ~ M submatrices. The transformation cost for ~ M is somewhat greater than for K ~ because of cancellation in the latter case that does not occur in the former. The cost of transforming damping matrices is comparable to that of transforming them for a global modal representation. Of course, when neither B nor? is used, or when damping is modeled using a linear combination of the mass and stiness matrices as a damping matrix, the cost of transforming the damping matrix is negligible. Overall, the cost of transforming the model can be expected to be several times the cost of a sparse factorization of the global stiness matrix. As a result of the transformation, the stiness matrix from the FE discretization is replaced with a much smaller diagonal matrix of substructure eigenvalues. The FE mass matrix is replaced with a much smaller matrix having ones on the diagonal. All other nonzeros can be conned to rectangular submatrices that represent coupling between substructures and their ancestors or descendants. Transformed damping matrices are similarly populated, except that the identity matrices on the diagonal are replaced with submatrices, which are diagonal in certain wellknown special cases. 3. OBTAINING SOLUTIONS Once the FE model has been transformed to the representation in terms of substructure eigenvectors, as described in the preceding section, it can be used to approximate either frequency response or the global eigensolution. The frequency response problem, after projection onto the substructure mode subspace, becomes 2?! M ~ + i! B ~ + (1 + i) K ~ + i? ~ p = ~ A(!)p = ~ F ; (5) where the dimensions of the matrices and the vector F ~ are reduced to the number of substructure modes computed. If many substructure modes have been computed, the matrix ~A(!) is very large, and it is sparse, but not diagonal. For this reason, solving the system of frequency response equations at every frequency is costly enough to motivate using a special frequency sweep algorithm.[2] In this algorithm, the frequency response is divided into two components. The rst component contains the response of approximate global modes whose natural frequencies lie in or very close to the frequency sweep interval, and this component accounts for resonant or nearly resonant behavior. The second component accounts for the remainder of the response and is represented in terms of the substructure modes. The portion of the response represented by the second component varies smoothly with frequency, so it can be approximated well by updating it only occasionally during the frequency sweep and extrapolating or interpolating for frequencies between updates. Whether for the rst component of the frequency response or for a global eigenanalysis, global eigenpairs can be approximated by solving the algebraic eigenvalue problem K ~ ~ = M ~ ~ ~ for ~ and. ~ Each column of ~ contains coecients of substructure eigenvectors for representing global eigenvectors, and ~ contains approximate global eigenvalues. In the transformation of the FE model to the substructure mode representation, substructure modes can be computed to very high cuto frequencies with only a nominal increase in computational cost. This increases the number of substructure modes and the accuracy of the frequency response analysis. However, since the dimension of the reduced eigenvalue problem increases, the cost of solving it increases signicantly. An approximate eigensolution can be obtained by including only rows and columns of K ~ and M ~ corresponding to substructure modes below a frequency higher than the highest frequency for which global modes are needed, but generally much lower than the substructure mode cuto frequency. This approximation can be improved considerably, and very economically, by means of iterations that access all of the computed substructure modes.[7] Once an eigensolution for the reduced problem has been obtained, the frequency sweep analysis begins at the lowest frequency of interest. The rst component of the response is computed, and an iterative scheme is used to obtain the second component. Typically, only one iteration, which mainly involves a matrix-vector multiply, is required to update the responses of the substructure modes for the second component of the frequency response. After computing the second component of the response for the rst frequency, it can typically be used for several frequencies before it needs to be updated again. The rst component of the frequency response is updated at each frequency, because it varies much less smoothly with frequency, and because it is inexpensive to compute since only a small number of global modes are involved. The second component of the frequency response is updated when the accuracy obtained by simply using what was computed for the rst response frequency is no longer acceptable. At this point, the old response becomes a starting vector for the iterative scheme. Once the update has been done, linear extrapolation can be used to represent the second component. When this 4

5 is no longer suciently accurate, another update is performed, which provides enough information for quadratic extrapolation. This pattern can be repeated throughout the frequency sweep analysis, enabling the use of a very large number of substructure modes without having to compute with them very often during the frequency sweep. If necessary, the results of a frequency sweep analysis or an eigenanalysis can be expanded from the substructure mode representation to all FE DOF. However, signicant savings in disk space and CPU time are possible if these results are only needed in selected DOF. One of the advantages of AMLS over the modal frequency response approach using the Lanczos eigensolver is that much less CPU time and disk space is committed to working with the full set of FE DOF. 4. NUMERICAL EXAMPLE For an example, AMLS is used to approximate the frequency response of a FE model of an automobile \body-in-white" (without doors, hood, trunk lid, or other \trim" items attached) which has about 2.9 million DOF. This model was created in industry for analysis using NASTRAN software. Frequency response results obtained using AMLS, on an IBM RS/6000 workstation, are compared to results obtained using modal frequency response, as well as direct frequency response, in MSC.NASTRAN on a Cray T90 vector supercomputer. A frequency sweep up to 400 Hz is performed, so global modes up to a cuto frequency of 600 Hz are calculated in MSC.NASTRAN, for modal frequency response using residual exibility. For AMLS, substructure eigenvectors with natural frequencies up to 4,000 Hz are computed and used to represent the structure. Global modes with natural frequencies up to 480 Hz are approximated in terms of these substructure eigenvectors for use in the rst component of the frequency response. In AMLS, because quasistatic dependence of eliminated DOF on forced DOF is represented throughout the model transformation process, the benet of residual exibility is obtained. In Phase 1 of the analysis process for using AMLS to nd frequency response, the stiness and mass matrices for the model are generated using NASTRAN and are written to les, along with data enabling AMLS software to refer to the user's grid point numbers, etc. In Phase 2, this FE model is automatically divided into 12,068 substructures on 24 levels based on the sparsity structure of the FE stiness and mass matrices. In Phase 3, for obtaining frequency response up to 400 Hz, a total of 40,336 substructure eigenpairs with natural frequencies up to 4,000 Hz are computed and the FE model is projected onto the substructure mode subspace. In Phase 4, an approximate eigensolution for the reduced model is obtained. The accuracy of the global eigenvalues found in this phase has a large impact on the accuracy of resonances in the frequency response. The rst exible mode natural fre- Table 1. Wallclock and CPU Times Wallclock Time AMLS on IBM Workstation CPU Time Phase 1 53:09 25:02 Phase 2 21:52 15:33 Phase 3 6:33:45 6:26:41 Phase 4 2:17:40 2:17:33 Phase 5 1:10:25 1:09:37 Total 11:16:51 10:34:26 Modal Frequency Response on Cray T90 Total 14:53:29 15:27:02 quency is only % higher than the value obtained by NAS- TRAN. Natural frequencies below 50 Hz are found with less than % error; below 100 Hz, with less than 0.015% error; below 200 Hz, with less than 0.048% error; below 300 Hz, with less than 0.16% error; and below 400 Hz, with less than 0.64% error. In Phase 5, the frequency sweep analysis is performed using the frequency sweep algorithm described in the preceding section. Wallclock and CPU times for the ve phases of the AMLS analysis are presented in Table 1, along with wallclock and CPU times for the modal frequency response analysis done in MSC.NASTRAN. The AMLS analysis was done on an IBM RS/ P Model 260 workstation which had two processors and only 1 gigabyte of RAM. However, only one processor was used for these results. The modal frequency response analysis was done on a Cray T90 supercomputer with 3 processors. All three processors were used at once at various times during the job, with the average number of processors in use concurrently, over the course of the job, equal to In the frequency response results from AMLS, responses at the drive point tend to be slightly more accurate than responses away from the drive point. Figure 2 shows a typical drive point inertance, with circles representing values computed using direct frequency response, a dashed line for modal frequency response computed using MSC.NASTRAN, and a solid line for frequency response computed using AMLS. The results are visually almost indistinguishable. Figure 3 shows a typical transfer inertance, again with circles, dashed line and solid line representing direct, MSC.NASTRAN modal, and AMLS frequency response respectively. In the transfer inertance, there is cancellation between responses of dierent global modes, so that the frequency response is about two orders of magnitude smaller than for the drive point inertance, and there are antiresonances which do not appear in the drive point case. 5

6 Inertance Frequency (Hz) Figure 2. Drive point inertance. (Circles: direct. Dashed: modal. Solid: AMLS.) Because of these antiresonances, the error in the AMLS approximations of natural frequencies is more apparent in transfer frequency response than in the smoother drive point response. The AMLS approximation of frequency response is shifted slightly to the right at high frequencies, and error in the shape of the frequency response curve in the antiresonances is noticeable. However, antiresonance behavior at high frequencies is seldom of engineering interest, and FE models are particularly unlikely to capture both natural frequencies and mode shapes accurately enough to represent antiresonances accurately. On this example, using AMLS on one workstation processor produces results of useful engineering accuracy in less elapsed time than was required for using modal frequency response with residual exibility on a Cray T90, even though multiple processors were used by the supercomputer. 5. CONCLUSIONS Until recently, frequency response analysis of automobile bodies has been very expensive, and limited in the range of frequencies that could be addressed. With the extremely large FE models needed for accuracy at higher frequencies, and the large number of response frequencies, the modal frequency response approach has been the only practical choice. However, the Lanczos eigensolution algorithm for computing natural frequencies and modes makes microprocessor-based computing platforms impractical because of their limited memory bandwidth. Vector supercomputers have become the standard computing platform, despite their high cost. The Automated Multi-Level Substructuring (AMLS) method is a new approach to large-scale frequency response analysis in which the FE model is automatically divided into thousands of substructures on many levels. Response is represented in terms of substructure eigenvectors rather than global modes of vibration. The numerical example in this paper demonstrates, on an automobile body model which has almost 3 million degrees of freedom, that it is faster to do frequency response analysis on a single workstation processor, using AMLS, than to use a multiprocessor Cray T90 supercomputer and the conventional modal frequency response approach. Despite the use of substructure eigenvectors rather than global eigenvectors, the accuracy obtained using AMLS is extremely good. In AMLS, improving accuracy by computing substructure eigenvectors to very high cuto frequencies is very inexpensive. The ability to use workstations for large-scale frequency response analysis, with the same job throughput that has been oered by supercomputers and very little reduction in accuracy, can be expected to have a dramatic impact on NVH computation in 6

7 10 1 Inertance Frequency (Hz) Figure 2. Transfer inertance. (Circles: direct. Dashed: modal. Solid: AMLS.) the automotive industry. In addition, because of the potential for parallelizing AMLS, the turnaround time for NVH analysis can be expected to decrease greatly in the near future. ACKNOWLEDGMENTS The modal frequency response analysis on the Cray T90 supercomputer was performed by Mr. Mladen K. Chargin. Supercomputer time was provided by SGI/Cray Research, Inc. The IBM RS/6000 workstation was provided by IBM Corporation under the Shared University Resource program. This research was funded by Ford Motor Company. REFERENCES 1. Bennighof, J. K. and Kaplan, M. F., \Frequency Window Implementation of Adaptive Multi-Level Substructuring," Journal of Vibration and Acoustics, Vol. 120, No. 2, pp , April Hurty, W. C., \Dynamic Analysis of Structural Systems Using Component Modes," AIAA Journal, Vol. 3, No. 4, pp , Craig, R. R., Jr. and Bampton, M. C. C., \Coupling of Substructures for Dynamic Analysis," AIAA Journal, Vol. 6, No. 7, pp , Du, I. S. and Reid, J. K., \The Multifrontal Solution of Indenite Sparse Symmetric Linear Systems," ACM Transactions on Mathematical Software, Vol. 9, pp , Du, I. S., Erisman, A. M. and Reid, J. K., Direct Methods for Sparse Matrices, Oxford University Press, New York, pp , Bennighof, J. K., Kaplan, M. F. and Muller, M. B., \Extending the Frequency Response Capabilities of Automated Multi-Level Substructuring," Proceedings of 41st AIAA/- ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference, Atlanta, April 2000 (in preparation). 2. Bennighof, J. K. and Kaplan, M. F., \Frequency Sweep Analysis using Multi-Level Substructuring, Global Modes and Iteration," Proceedings of 39th AIAA/ASME/ASCE/- AHS Structures, Structural Dynamics and Materials Conference, Long Beach, April