Browsing by Subject "Frequency response (Dynamics)"
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Item An efficient eigensolution method and its implementation for large structural systems(2004) Kim, Mintae; Bennighof, Jeffrey KentThe automated multilevel substructuring (AMLS) method, which was originally designed for efficient frequency response analysis, has emerged as an alternative to the shift-invert block Lanczos method [23] for very large finite element (FE) model eigenproblems. In AMLS, a FE model of a structure, typically having a ten million degrees of freedom, is automatically and recursively divided into more than ten thousand substructures on dozens of levels. This FE model is projected onto the substructure eigenvector subspace which typically has a dimension of 100,000. Solving the reduced eigenproblem on the substructure eigenvector subspace, however, is unmanageable for modally dense models which typically contain more than 10,000 eigenpairs. In this dissertation, a new eigensolution algorithm for the reduced eigenproblem produced by the AMLS transformation is presented for large structural systems with many eigenpairs. The new eigensolver in combination with AMLS is advantageous for solving the eigenproblems for huge FE models with many eigenpairs because it takes much less computer time and resource than any other existing eigensolvers while maintaining acceptable eigensolution accuracy. Therefore, the new eigensolution algorithm not only makes high frequency analysis possible with acceptable accuracy, but also extends the capability of solving large scale eigenvalue problems requiring many eigenpairs. A reduced eigenvalue problem produced by the AMLS transformation for a large finite element model is defined on the substructure eigenvector subspace. A new distilled subspace is obtained by defining subtrees in the substructure tree, solving subtree eigenproblems, and truncating subtree and branch substructure eigenspaces. Then the reduced eigenvalue problem on the substructure eigenvector subspace is projected onto the smaller distilled subspace, utilizing the sparsity of the stiffness and mass matrices. Using a good guess of a starting subspace on the distilled subspace, which is represented by a sparse matrix, one subspace iteration recovers as much accuracy as needed. Hence, the size of the eigenvalue problem for Rayleigh-Ritz analysis can be greatly minimized. Approximate global eigenvalues are obtained by solving the Rayleigh-Ritz eigenproblem on the refined subspace, computed by one subspace iteration, and the corresponding eigenvectors are recovered by simple matrix-matrix multiplications. For robustness of the implementation of the new eigensolution algorithm, the remedies for a nearly singular stiffness matrix and an indefinite mass matrix are presented. Also, the new eigensolution algorithm is very parallelizable. The parallel implementation of this new eigensolution algorithm for shared memory multiprocessor machines is done by using OpenMP Application Program Interface (API) for performance improvement. Timing and eigensolution accuracy of the implementation of the new eigensolution algorithm are presented, compared with the results from the block Lanczos eigensolver in the commercial software MSC.Nastran. In addition to the new eigensolution algorithm, a new method for an augmented eigenproblem for residual flexibility is developed to mitigate loss of accuracy by paying little computational cost in modal frequency response analysis.Item Efficient frequency response analysis of structures with viscoelastic materials(2006) Swenson, Eric Dexter; Bennighof, Jeffrey K.Item Frequency response computation for complex structures with damping and acoustic fluid(2004-12) Kim, Chang-wan, 1969-; Bennighof, Jeffrey Kent, 1960-Modal frequency response analysis is a very economical approach for large and complex structural systems since there is an enormous reduction in dimension from the original finite element frequency response problem to the number of modes participating in the response. When damping does not exist, the modal frequency response problem is inexpensive to solve because it becomes uncoupled. However, when damping exists, the modal damping matrices can become fully populated, making the modal frequency response problem expensive to solve at many frequencies. The conventional approach to solve the modal frequency response problem with damping is to use either direct methods with O(n 3 ) operations at each frequency, or iterative methods with O(n 2 ) operations per iteration and numerous iterations at each frequency, where n is the number of modes used to represent the response. Another approach is to use a state space formulation and an eigensolution to uncouple the damped modal frequency response problem, but this doubles the dimension of the problem. All of the existing traditional methods are very expensive for systems with many modes. In this dissertation, a new algorithm to solve the modal frequency response problem for large and complex structural systems with structural and viscous damping is presented. The newly developed algorithm, fast frequency response analysis (FFRA), solves the damped modal frequency response problem with O(n 2 ) operations at each frequency. The FFRA algorithm considers both structural damping and viscous damping for structural systems. When only structural damping exists, the modal frequency response problem is uncoupled by applying the eigensolution of the complex symmetric modal stiffness matrix. A complex symmetric matrix eigensolver (CSYMM) has been developed to solve the complex symmetric matrix eigenvalue problem efficiently. If a viscous damping matrix is also present, the algorithm handles viscous damping by noting that the rank of the viscous damping matrix is typically very low for the problems of interest in the automobile industry because of the small number of viscous damping elements such as shock absorbers and engine mounts. This algorithm has also been applied to the coupled response of systems consisting of a light acoustic fluid and structure, and systems with enforced motion. Also, the algorithm is implemented in parallel on shared memory multiprocessor machines for performance improvement. The FFRA algorithm is evaluated for several industry finite element models which have millions of degrees of freedom. The FFRA algorithm produces outstanding performance compared to the methods available in the commercial finite element software MSC.Nastran or NX.Nastran in terms of analysis time, since the new algorithm is many times faster while obtaining almost the same accuracy as MSC.Nastran. Therefore, the new FFRA algorithm makes inexpensive high frequency analysis possible and extends the capability of solving modal frequency response analysis to higher frequencies.Item Implementation of automated multilevel substructuring for frequency response analysis of structures(2001-12) Kaplan, Matthew Frederick; Bennighof, Jeffrey Kent, 1960-In the design of vehicles, such as automobiles, aircraft, spacecraft, or submarines, it is important to be able to accurately predict dynamic behavior of the structure. With the extremely high cost of building physical prototypes of these vehicles, there is a growing emphasis on analysis of computer models. In this dissertation, a method known as Automated Multilevel Substructuring (AMLS) is presented for accurately solving frequency response problems involving large, complex models with millions of degrees of freedom. Conventional methods for addressing these problems, such as mode superposition using a Lanczos eigensolver or model reduction using component mode synthesis, are reviewed. The Automated Multilevel Substructuring (AMLS) method partitions finite element models into substructures, similar to component mode synthesis methods, but uses an automated partitioning procedure that reduces the burden on the analyst. The finite element matrices are projected onto a reduced subspace, on which the frequency response is computed. Two frequency response algorithms are presented. Both methods require the solution of a global eigenvalue problem on the reduced subspace. The first method uses straightforward mode superposition. The second method employs a new iterative approach in which the modal frequency response leads to a residual problem that is solved using an iterative splitting method. The global eigensolution and frequency response algorithms are specifically designed to take advantage of the properties of the reduced subspace. Numerical examples are presented for models with millions of degrees of freedom. The performance and accuracy of the AMLS method are compared to the standard commercial software package for large-scale linear dynamic analysis. These examples establish that AMLS can be used to accurately obtain the response of very large models with significantly less computational resources than competing methods. In comparison to the modal frequency response obtained with the standard commercial software package using a shifted block Lanczos algorithm, AMLS ran up to 6.4 times faster, used less memory, and required an order of magnitude less data transfer. Thus, the AMLS method makes it possible to do frequency response analysis of large, complex structures at higher frequencies than was previously practical.