42 Robust Expansion of Experimental Mode Shapes Under Epistemic Uncertainties 421 42.2 Robust ECRE-Based Expansion 42.2.1 ECRE-Based Expansion: Formulation In this article, a brief description of CRE concept is proposed. The reader can refer to [16–18] for a more in-depth description of CRE concepts. For the sake of clarity, we will consider that the model is erroneous only in term of stiffness. Let .fug; fvg/, two displacement fields such as: • fug is kinematically admissible, i.e., verifies the equilibrium equations; • fvg is derived from the constitutive relation between the stress and the displacement such as the fieldfu vg expresses the error in stiffness in the model. The FE discretization yields to the following energy-based functional: 2 ! .fug; fvg/ Dfu vg T ŒK fu vg: (42.2) An extended formulation is proposed, namely ECRE [12], and includes test data fQug: e2 ! .fug; fvg/ D 2 ! .fug; fvg/ C r 1 r f …u QugT ŒKR f…u Qug; (42.3) with Œ… , a projection operator from the FE model space to the observation space, ŒKR , the stiffness matrix of the nominal model reduced to the measurement dofs2 and r 2 Œ0I 1 a weighting coefficient allowing the relative confidence in the identified eigenvectors to be taken into account in the cost function e2 !. The idea behind CRE concepts is to minimize the non-reliable data while relaxing admissibility constraints: minimize e2 ! under the constraint ŒK fu vg ŒK !2 ŒM fugD0: (42.4) To solve the problem (42.4), the constraint equations will be introduced via a vector of Lagrange multipliers f g. The optimization problem thus becomes: min˚f .fug; fvg; f g/ De2 ! Cf g T ŒK fu vg ŒK !2 ŒM fug : (42.5) Equation (42.5) is a saddle-point problem: the solution is defined by the stationarity conditions of the functionf with respect to the unknown vector fug,fvg andf g: @f @fug D0 @f @fvg D0 @f @f g D0: (42.6) Rearranging equations and using the fact that for any hermitian matrix ŒM and for any fxg 2 CN, @fxgŒM fxg @fxg D 2ŒM fxg, Eq. (42.6) yields to the following linear system: ŒA fxgDfbg; (42.7) where: ŒA D ŒK ŒK !2 ŒM ŒK !2 ŒM r 1 r Œ… T ŒKR Œ… f xgD u v u and fbgD f 0g r 1 r Œ… T ŒKR fQug : The ECRE-based expansion involves the solution of a linear system in order to obtain the unmeasured part of the identified eigensolutions, denotedfug. 2In practice, the Guyan stiffness matrix is often used.
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