Reduced-Order Design-Oriented Stress Analysis Using Combined Direct and Adjoint Solutions

Abstract

A new method for extracting accurate stress information from reduced-order structural and aeroelastic models is presented. The method has second-order accuracy when approximate reduced-order direct and adjoint solutions (based on different reduced-order bases) are used simultaneously to obtain approximate stresses. The method is applicable to both static and dynamiclinear analysis. A review of four common methods for structural model order reduction [two variants of the mode displacement method (standard mode displacement and the ctitious mass method), the mode acceleration method, and the Ritz vector method] identi es sources of dif culty and causes of errors in stress behavior sensitivity calculations. Considerations used for selection of the reduced-order direct and adjoint bases are discussed. A series of static and dynamic test cases is used to assess accuracy of the new method in an analysis mode. Accuracy studies of sensitivity calculations follow. We hope to contribute to the eld of design-oriented structural dynamics in terms of both insight and practice.

Optimal modal reduction of vibrating substructures

“…There is a very large volume of literature on the subject. See references in the review papers [ [2][3][4][5][6]. In addition, books and conference proceedings on robust control (see, e.g., References [7,8]) often include chapters on model reduction.…”

Which are the important modes of a subsystem?

“…There is a very large volume of literature on the subject, often called 'model order reduction;' see, e.g. the review papers [1][2][3][4][5]. Some of this work is related to robust control, where there is a need for the repetitive real-time solution of large structural dynamic problems [6].…”

Optimal modal reduction of dynamic subsystems: Extensions and improvements

“…The eigenvectors are orthogonal, and are normalized with respect to the mass matrix [18]. Let U be the N s × N s matrix whose columns are the eigenvectors / n , and X 2 be the N s × N s diagonal matrix whose diagonal entries are the squared frequencies 2 n . Now, the third set of equations in (11) describes the dynamics of 's':…”

“…Two areas in which the role of model reduction methods has been prominent are structural dynamics, which motivates the current study, and real-time robust control [1]. There is a broad literature on this subject; see, e.g., the review papers [2][3][4][5], and the recent papers [6][7][8][9][10] in the computational mechanics literature.…”

“…• Let I r be the N r × N r identity, X 2 r be the N r × N r diagonal matrix whose diagonal entries are the squared frequencies 2 n of the first N r modes and U r be the N s × N r matrix whose columns are the first N r eigenvectors / n . Then…”