Page 189 - Numerical Analysis and Modelling in Geomechanics
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170 ANNAMARIA CIVIDINI AND GIANCARLO GIODA
(6.3)
where vector collects all the measured displacements, and f 1 and f 2 are known
nodal force vectors.
A static condensation of eq.(6.3) leads to,
(6.4)
where
(6.5)
Eq. (6.4), by taking into account eq. (6.2), can be written in the following form,
(6.6)
where
(6.7)
The stiffness matrices in eq. (6.7) are obtained by partitioning matrix K . with the
i
same criteria used in eq. (6.3) for matrix K.
Grouping the unknown parameters in the 2n vector p, and grouping vectors r i
in the m×2n matrix R,
(6.8)
eq. (6.6) yields the following relationship that governs the back analysis problem,
(6.9)
Assuming that the number of measured displacements exceeds the number of
unknown elastic constants, a standard least square minimisation can be applied to
eq. (6.9), which leads to the following non-linear equation system,
(6.10)
The non-linear nature of eq. (6.10) derives from the fact that the matrix of
coefficients R depends, through matrix Q, on the unknown vector p.
