Abstract
The finite element model errors mainly come from the complex parts of the geometry, boundary conditions and stress state of the structure. Therefore, the problem for updating mass and stiffness matrices can be reduced to an inverse problem for symmetric matrices with submatrix constraints (IP-MUP): Let and
be the measured eigenvalue and eigenvector matrices with rank(Φ) = p. Find n × n symmetric matrices M and K such that
where M(r) and K(r) are the r × r leading principal submatrices of M and K, respectively. We then consider an optimal approximation problem (OAP): Given n × n symmetric matrices Ma and Ka. Find
such that
, where
is the solution set of Problem IP-MUP. In this paper, the solvability condition for Problem IP-MUP is established, and the expression of the general solution of Problem IP-MUP is derived. Also, we show that the optimal approximation solution
is unique and derive an explicit formula for it.
Acknowledgments
The authors would like to express their gratitude to the anonymous referees and Professor Federico Poloni (Handling Editor) for their valuable suggestions and comments that improved the presentation of this manuscript.
Disclosure statement
No potential conflict of interest was reported by the authors.