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

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 Λ = d i a g (λ 1 , ... , λ p) ∈ R p × p and Φ = [ ϕ 1 , ... , ϕ p ] ∈ R n × p be the measured eigenvalue and eigenvector matrices with rank(Φ) = p. Find n × n symmetric matrices M and K such that K Φ = M Φ Λ , Φ ⊤ M Φ = I p , s. t. M (r) = M 0 , K (r) = 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 Ma and Ka. Find (M ˆ , K ˆ) ∈ S E such that ∥ K ˆ − K a ∥ 2 + ∥ M ˆ − M a ∥ 2 = min (M , K) ∈ S E (∥ K − K a ∥ 2 + ∥ M − M a ∥ 2) , where 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 (M ˆ , K ˆ) is unique and derive an explicit formula for it. [ABSTRACT FROM AUTHOR]