A direct method for updating mass and stiffness matrices with submatrix constraints

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 $\mathrm{\Lambda }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\lambda }_1,\dots ,{\lambda }_p)\in {\mathbb{R}}^{p\times p}$ and $\mathrm{\Phi }=[{\varphi }_1,\dots ,{\varphi }_p]\in {\mathbb{R}}^{n\times p}$ be the measured eigenvalue and eigenvector matrices with rank(Φ) = p. Find n × n symmetric matrices M and K such that $K\mathrm{\Phi }=M\mathrm{\Phi }\mathrm{\Lambda },{\mathrm{\Phi }}^{\mathrm{\top }}M\mathrm{\Phi }=I_p,\mathrm{s}.\mathrm{t}.M_{\left(r\right)}=M_0,K_{\left(r\right)}=K_0,$ 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 M_a and K_a. Find $(\hat{M},\hat{K})\in {\mathcal{S}}_E$ such that $\parallel \hat{K}-K_a{\parallel }^2+\parallel \hat{M}-M_a{\parallel }^2=\underset{(M,K)\in {\mathcal{S}}_E}{min}(\parallel K-K_a{\parallel }^2+\parallel M-M_a{\parallel }^2)$, where ${\mathcal{S}}_E$ 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 $(\hat{M},\hat{K})$ is unique and derive an explicit formula for it.

Keywords:

• Finite element model updating
• submatrix constraint
• optimal approximation

AMS Classifications:

• 65F18
• 15A24

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.