smart technologies for safety engineering

VDM-Based Health Monitoring 79
water head H will be expressed as
HK = H L K + H R K
Rd
+ HK = H L K + DHKjβ0j + DHd Kjε0j In the numerical algorithm the distortion β 0 has to be determined (first with ε 0 = 0asa
solution of the following system of equations, including only the network branches of the
nonlinear constitutive characteristics (cf. Equation (64)):
(1 − γi)Ni K D H Kj β0 j =−(1 − γi)(ε L i − ˜εi) − (1 − γi)Ni K D Hd
Kj ε0 j
Having determined β 0 , the primary design variable ε 0 , which models leakage in the network,
can be calculated by TOLMIN. If the detected leakages have an influence on the distribution of
flow in the branches of the nonlinear characteristics, the resultant ε 0 is substituted to Equation
(81) and a new β 0 is found. The iterative process proceeds until convergence is obtained, i.e.
the final ε 0 does not imply any variation of β 0 .
3.4.4.4 Sensitivity Analysis
As demonstrated previously, leakages in the water network can be simulated through a virtual
distortion state. The original (linear) response HL of the network is combined with the residual
response HR , generated by virtual distortions, in order to match the measured response
HM . The components of virtual distortions (design variables) are calculated by the quadratic
programming routine TOLMIN and therefore a sensitivity analysis allowing for the calculation
of gradients is necessary.
Having an analytical description of water network relations, presented in Section 3.4.2, the
gradient of the objective function (76) for linear constitutive relations can be calculated in the
following way:
∂ F
∂ε 0 j
=
2
M HK 2
L
HK + D Hd
Kjε0j − H M Hd
K DKj In the case of the nonlinear constitutive relation advantage can be taken of the following chain
differentiation (cf. Equation (55) in Chapter 2):
∂ F ∂ F ∂H ∂H
= + 0 ∂ε ∂H 0 ∂ε ∂β0 ∂β0 ∂ε0
(83)
The components ∂H/∂ε 0 and ∂H/∂β 0 are determined from Equation (80), whereas the component
∂β 0 /∂ε 0 is calculated from Equation (81).
3.4.5 Numerical Examples
3.4.5.1 Example 1
Assume now that the water head distribution H M (cf. Equation (73)) has been measured in every
node of the water network shown in Figure 3.39, which may be subject to leakages. Location as
well as intensity of the leakages needs to be identified, assuming that they may be located in any
plastic
(80)
(81)
(82)