Dynamic analysis of complex two- and three-dimensional structures frequently involves finite element discretizations with over a million unknowns. The discretization is typically used for frequency response at many frequencies so that a dimensional reduction step is advantageous or even necessary. The standard reduction approach is modal truncation, which requires a costly partial eigensolution but reduces the number of unknowns by orders of magnitude.

While modal truncation is justified in the continuous setting-higher eigenfunctions have much lower participation in the response than lower ones-there is another important reason. The modes that are well approximated by the finite element discretization are the only modes that should be retained. Hence the cost of the frequency response is reduced greatly without a significant loss of accuracy. The cost of the partial eigensolution required for modal truncation increases dramatically as the frequency range for the analysis increases because the number of eigenpairs needed can easily reach into the thousands. High modal density (close spacing of eigenvalues) also contributes to the cost. An alternative to this approach is the Automated Multi-Level Substructuring (AMLS) method in which the structure is recursively divided into thousands of subdomains. Eigenvectors associated with these subdomains are used to represent the structure's response rather than the traditional global eigenvectors. Reduction of the finite element discretization is based on many small, local, inexpensive eigenvalue problems.

This presentation examines the mathematical basis for AMLS in the continuous variational setting. Differential eigenvalue problems are defined on subdomains and on interfaces between subdomains. For the interface eigenvalue problems, an operator is defined that acts on interface trace functions and consistently represents mass associated

with extensions of those trace functions. With this new operator, all of the differential eigenvalue problems are projections of the global eigenvalue problem onto a hierarchy of mutually orthogonal subspaces.

The objective of our dimensional reduction is to retain only as many eigenfunctions as are needed so that the eigenspace truncation error is consistent with the finite element discretization error. AMLS is an alternative to conventional approaches for solving large systems of equations arising from a finite element discretization: instead of

approximating the solution of a system of equations, identify the subspace that contains good approximate solutions of the partial differential equation and project the problem onto that subspace. This approach is advantageous particularly when many solutions are needed from one discretization.

(Joint work with Jeff Bennighof, University of Texas, Aerospace and Engineering Mechanics)