Efficient frequency response computation for structures with structural damping




Palmer, Jeremiah Fletcher

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The modern procedure for analyzing the dynamics of a large, complex structure, such as an automobile, is to use the finite element method to discretize the structure with millions of degrees of freedom. For the steady-state response to a harmonic excitation, a frequency response problem (FRP) is derived for the finite element discretization. To ease computational cost, modal analysis is performed, creating a corresponding FRP in an approximating modal subspace with a substantial reduction in dimension. Typically, more than one level of structural damping is present in a complex structure. This results in a fully populated modal damping matrix, so that the frequency-dependent coefficient matrix of the modal FRP is full. This problem is traditionally solved using a brute-force approach, which can be prohibitively expensive since it requires O(n³) operations for each of the hundreds of frequencies. This dissertation presents two new approaches for solving modal FRPs of automobile structures that have any composition of structural damping. Each approach requires a single frequency-independent O(n³) operation which changes the full coefficient matrix of the modal FRP into one with a simpler form. The first approach presents a new method which creates a low rank approximation of the modal structural damping matrix. The second approach is used when the modal structural damping matrix has high rank and relies on a new method for determining an accurate eigenvalue decomposition of a complex symmetric matrix. Computing responses using these two approaches then only requires O(n²) operations for every frequency. Automobile companies perform analyses on computers with multi-core CPU processors and graphics processing units which can perform dense linear algebra operations with high efficiency. This dissertation shows how the two approaches are implemented to take advantage of these parallel technologies. The accuracy and performance of the two new approaches are presented and compared with the brute-force approach


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