## 3 Error Estimation for a Reduced Flexible Body
Replacing \(\chi_{n}^{i}\) with \(\overline{\chi}_{n}^{i}\) in the right-hand side of Equation (31), we get:  
\[\zeta_{n}^{i}=\Xi_{n}^{i}{}^{T}\left[2\overline{\chi}_{n}^{i}\overline{ \mathbf{T}}_{0}^{i}{}^{T}\mathbf{M}_{F}^{i}\overline{\mathbf{T}}_{r}^{i}-2 \overline{\mathbf{T}}_{0}^{i}{}^{T}\mathbf{K}_{F}^{i}\overline{\mathbf{T}}_{r }^{i}+[\overline{\chi}_{n}^{i}]^{2}\overline{\mathbf{T}}_{r}^{i}{}^{T}\mathbf{ M}_{F}^{i}\overline{\mathbf{T}}_{r}^{i}-\overline{\chi}_{n}^{i}\overline{\mathbf{T}}_{r}^{i} \mathbf{K}_{F}^{i}\overline{\mathbf{T}}_{r}^{i}\right]\Xi_{n}^{i}, \tag{33}\]  
and the component matrices in Equation (33) are defined by:  
\[\overline{\mathbf{T}}_{0}^{i}{}^{T}\mathbf{M}_{F}^{i}\overline{\mathbf{T}}_{r }^{i}= \begin{bmatrix}\mathbf{0}&\mathbf{X}_{a}^{i}\\ \mathbf{0}&\mathbf{X}_{b}^{i}\end{bmatrix} ,\ \ \overline{\mathbf{T}}_{r}^{i}{}^{T} \mathbf{K}_{F}^{i}\overline{\mathbf{T}}_{r}^{i}= \begin{bmatrix}\mathbf{0}& \mathbf{0}\\ \mathbf{0}&\mathbf{X}_{b}^{i}\end{bmatrix} ,\ \ \mathbf{X}_{a}^{i}=\mathbf{\Phi}_{d}^{i}{}^{T} \mathbf{M}_{s}^{i}\mathbf{F}_{rs}^{i}\mathbf{A}^{i},\ \ \mathbf{X}_{b}^{i}=\mathbf{A}^{i}{}^{T} \mathbf{F}_{rs}^{i}\mathbf{A}^{i}, \tag{34a}\] \[\overline{\mathbf{T}}_{0}^{i}{}^{T}\mathbf{K}_{F}^{i}\overline{\mathbf{T}}_{r }^{i}= \begin{bmatrix}\mathbf{0}&\mathbf{Y}_{a}^{i}\\ \mathbf{0}&\mathbf{Y}_{b}^{i}\end{bmatrix} ,\ \ \mathbf{Y}_{a}^{i}=\mathbf{\Phi}_{d}^{i}{}^{T} \mathbf{K}_{s}^{i}\mathbf{F}_{rs}^{i}\mathbf{A}^{i},\ \ \mathbf{Y}_{b}^{i}=[\mathbf{\Psi}^{i}{}^{T} \mathbf{K}_{s}^{i}+\mathbf{K}_{c}^{i}]\mathbf{F}_{rs}^{i}\mathbf{A}^{i},\] (34b) \[\overline{\mathbf{T}}_{r}^{i}{}^{T}\mathbf{M}_{F}^{i}\overline{\mathbf{T}}_{r }^{i}= \begin{bmatrix}\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{A}^{i}{}^{T}\mathbf{\Phi}_{r}^{i}[\mathbf{\Lambda}_{r}^{i} ]^{-2}\mathbf{\Phi}_{r}^{i}{}^{T}\mathbf{A}^{i}\end{bmatrix} . \tag{34c}\]  
Using the definition of the constraint mode \(\mathbf{\Psi}^{i}\) in Equation (13) and the orthogonality of the eigenvector matrices \(\mathbf{\Phi}_{d}^{i}\) and \(\mathbf{\Phi}_{r}^{i}\) yields:  
\[\mathbf{X}_{a}^{i}=\mathbf{\Phi}_{d}^{i}\mathbf{M}_{s}^{i}\mathbf{\Phi}_{r}^{ i}[\mathbf{\Lambda}_{r}^{i}]^{-1}\mathbf{\Phi}_{r}^{i}{}^{T}\mathbf{A}^{i}= \mathbf{0}, \tag{35a}\] \[\mathbf{Y}_{a}^{i}=\mathbf{\Phi}_{d}^{i}\mathbf{K}_{s}^{i}\mathbf{\Phi}_{r}^{ i}[\mathbf{\Lambda}_{r}^{i}]^{-1}\mathbf{\Phi}_{r}^{i}{}^{T}\mathbf{A}^{i}= \mathbf{0},\] (35b) \[\mathbf{Y}_{b}^{i}= \begin{bmatrix}-\mathbf{K}_{c}^{i}[\mathbf{K}_{s}^{i}]^{-1} \mathbf{K}_{s}^{i}+\mathbf{K}_{c}^{i}{}^{T}\end{bmatrix} \mathbf{F}_{rs}^{i} \mathbf{A}^{i}=\mathbf{0}. \tag{35c}\]  
Substituting Equations (35a), (35b), and (35c) into Equations (34a) and (34b), Equation (33) can be rewritten using Equations (34a) and (34c) as:  
\[\zeta_{n}^{i}=\Xi_{n}^{i}{}^{T}\left(\overline{\chi}_{n}^{i} \begin{bmatrix} \mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{X}_{b}^{i}\end{bmatrix} +[\overline{\chi}_{n}^{i}]^{2} \begin{bmatrix}\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{A}^{i}{}^{T}\mathbf{\Phi}_{r}^{i}[\mathbf{\Lambda}_{r}^{i}]^{- 2}\mathbf{\Phi}_{r}^{i}{}^{T}\mathbf{A}^{i}\end{bmatrix} \right)\Xi_{n}^{i}. \tag{36}\]  
In the previous work [23], the second term of the right-hand side of Equation (36) was neglected for computational efficiency of the error estimator, but it causes the predicted error to be at the low bound of the exact error. To handle this problem, a numerical treatment is considered in this work. In Equation (34c), \(\mathbf{\Lambda}_{r}^{i}\) is a diagonal matrix that includes residual eigenvalues that are larger than the maximum dominant eigenvalue. When the eigenvalues are sorted in ascending order, the following inequality equation with the matrix norm is obtained:\[\frac{1}{\kappa_{N_{d}+1}^{i}}\|\mathbf{F}_{rs}^{i}\|<\frac{1}{\kappa_{n}^{i}}\| \mathbf{F}_{rs}^{i}\|, \tag{37}\]  
where \(\kappa_{N_{d}+1}^{i}\) is the first residual eigenvalue and \(\kappa_{n}^{i}\) is among the dominant eigenvalues (\(n\leq N_{d}<N_{d}+1\)).