iFEM	Inverse finite element method
CFTSC	Coarse and fine two-stage calibration
SCFN	Self-constructed fuzzy networks
PSO	Swarm optimization algorithm
SOPSO	Single-objective particle swarm optimization algorithm
NURBS	Non Uniform Rational B-spline
DOF	Degrees of freedom
MCMC	Markov Chain Monte Carlo
RMSE	Root mean square error
RRMSE	Relative root mean square error
NDI	Northern digital incorporated
FBG	Fiber bragg grating
List of Symbols
$e(u)$	Theoretical sectional strain
$e^{\epsilon }$	Actual sectional strain
$d^t$	The displacement at any position of the section
$H(x)$	Shape function
$u_e$	Nodal degrees of freedom
$e_u$	The strain at any position of the cross-section
M(x)	Strain function matrix
$k^e$	Elements stiffness matrix
$f^{\mu }$	Elements stiffness vector
l	The unit length after reconstruction
n	The number of section strains on the neutral axis
$x_i$	The location of the calculated section strain.
$w_p$	Weighting factor
${\epsilon }^a$	Surface strains
$x_i$	Sensor axial coordinates
${\theta }_i$	Sensor circumference angle
${\beta }_i$	Angle between sensor direction and axial direction
$T$	Conversion relationship between surface strain and section strain
$\mu$	Poisson ratio
$r$	The outer radius of the section
$D_y$	Bending stiffness in y-direction
$G_z$	Shear stiffness in z-direction
$T_k$	Displacement–strain transformation matrix
u,v,x	Displacement along each axis
θx, θy, θz	Rotation angles of each axis
$d$	The displacement of the final reconstruction
$T_k^{\prime }$	Actual displacement–strain transformation matrix
$dis{p}_i^u$	actual measured displacement at check points
$dis{p}_i^v$	Theoretical displacements reconstructed by iFEM
n	The number of position sensors (check points)
P′	The actual installation positions of strain sensors
$d_i^t$	The theoretical displacement of the iFEM reconstruction
$d_i^a$	Actual displacement
$\Delta d_i$	Reconstruction error
$G$	Displacement field matrix
$N_i$	Shape function
$\Delta u$	Nodal degrees of freedom error
$\beta$	The residuals after displacement error correction
$p(\Delta u,{\sigma }^2)$	Joint prior distribution
N(Δuj,$\Delta s_j^2$)	The conditional posterior distribution of Δu
$Q_j$	The jth sample data under working conditions
${\epsilon }_l$	The lth strain value measured by the sensor
$\Delta u_i$	The kinematic variable in the nodal DOF error
Fi	Decompose the original sample data Qj
$k_i$	Weight factors
$M_i$	Control vertex
$E_{i,c}(t)$	The c-th order B-spline basis function
$t_i$	knots
T	Nodal vector
$\widehat{F}$ i	Fitted data
${\widehat{\epsilon }}_{\mathrm{l}}$	Fitted strain
$\Delta {\widehat{u}}_{\mathrm{i}}$	Fitted kinematic variables
ai	The weighting constants corresponding to the kinematic variable error
$H_{ki}$	Transformation matrix
et	The integrated residual value of sample under one working condition
$F^{sc}$	Screened data set
$S^n$	The nth fuzzy rule
${\epsilon }_{\mathrm{k}}$	Input values for fuzzy self-configured networks
${\widehat{p}}^n$	Output of the nth fuzzy rule
$w^n$	The value corresponding to fuzzy rule output
$q_n$	The weighting constants of the rule
${\eta }_t$	The error standard of SCFN
ηa	Predetermined error thresholds
$\widehat{U}(c)$	Actual output of SCFN
$U(c)$	Desired output of SCFN
$R_k^n$	Membership degree
$a^n(i)$	The rule adjustment result at moment i
V	Adjust the adaptive speed during rule adjustment
${\phi }^n(i-1)$	The weighting constants of the nth rule at moment i − 1