We present an automated multilevel substructuring (AMLS) method for eigenvalue computations in linear elastodynamics in a variational and algebraic setting. AMLS first recursively partitions the domain of the PDE into a hierarchy of subdomains. Then AMLS recursively generates a subspace for approximating the eigenvectors associated with the smallest eigenvalues by computing partial eigensolutions associated with the subdomains and the interfaces between them. We remark that although we present AMLS for linear elastodynamics, our formulation is abstract and applies to generic H 1 -elliptic bilinear forms.In the variational formulation, we define an interface mass operator that is consistent with the treatment of elastic properties by the familiar Steklov-Poincaré operator. With this interface mass operator, all of the subdomain and interface eigenvalue problems in AMLS become orthogonal projections of the global eigenvalue problem onto a hierarchy of subspaces. Convergence of AMLS is determined in the continuous setting by the truncation of these eigenspaces, independent of other discretization schemes.The goal of AMLS, in the algebraic setting, is to achieve a high level of dimensional reduction, locally and inexpensively, while balancing the errors associated with truncation and the finite element discretization. This is accomplished by matching the mesh-independent AMLS truncation error with the finite element discretization error. Our report ends with numerical experiments that demonstrate the effectiveness of AMLS on a model problem and an industrial problem.

An algorithm is presented for performing frequency sweep analysis on large finite element models transformed by Automated Multi-Level Substructuring. As a result of the transformation, response is represented in terms of substructure modes of vibration. In the frequency sweep algorithm, frequency response is represented in terms of two components. The first component is in terms of global modes of vibration, which can be obtained very economically from the transformed model, and the second component represents the remainder of the response. Because all global modes that are near resonance are included in the first component, the second component varies smoothly with frequency, so that it can be approximated very effectively with extrapolation. A numerical example demonstrates the efficiency and accuracy of the algorithm.

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Adaptive multi-level substructuring (AMLS) is a method for reducing the order of a complex structure’s finite element model by orders of magnitude, while ensuring that the accuracy available from the original model is preserved. A structure’s finite element model is transformed to a much more efficient representation in terms of approximate vibration modes for substructures on multiple levels. An adaptive procedure constructs an optimal model for satisfying a user-specified error tolerance, by determining which modes should be included in the model. In this paper, a frequency window implementation of AMLS is developed, in which frequency response analysis can be done over a frequency window at little additional cost beyond that of the center frequency solution. A numerical example is presented.

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