PSI - Issue 5

Mikhail Tashkinov et al. / Procedia Structural Integrity 5 (2017) 577–583 Mikhail Tashkinov et al. / Structural Integrity Procedia 00 (2017) 000 – 000

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structure or structure-like element (hereinafter referred to as a sample) with embedded fiber optics with Bragg gratings as well as corresponding per-layer finite element model. The positions of the Bragg gratings in the sample determine the control points and should be reliably noted, for example, using X-ray or other methods of nondestructive testing. In the case of a quasi-isotropic material, which is characterized by two independent constants, one sensor line is sufficient. The mesh of the finite-element model is constructed in such a way that the geometric location of the Bragg gratings corresponds to specific mesh nodes. The properties of the material layer known to a certain accuracy are specified, which are the initial set of constants. Next, the sample is being loaded with force or displacement control, the same boundary conditions are also specified for the finite element model. During deformation of the sample, the FBG sensor readings are recorded at the points, thus dependence of deformations on the values of the applied load is being created. Similar data is extracted from the simulation results. Based on the difference in the values of the experimentally measured deformation and values calculated at the control points using finite element model, a minmization functional is constructed. So, it is possible to find material constants values that would correlate the calculation results to the deformations recorded by the FBG sensors at a given load. The material constants become optimization parameters, that, in the final, should ensure the simulation results and the FBG readings are consistent in the control points. To optimize the parameters, a regression model is created and the Levenberg-Marquardt (L-M) algorithm is used, which consists of successively approximations of the given initial values of the parameters to the sought local optimum (Levenberg (1944), Marquardt (1963)). The regression sample set is ( , ), where are the strain values at the reference point under load F . The regression function is the constitutive relation for isotropic media that connects stresses and strains. So, if the optical fiber is aligned to the axis 1, then the function is written as: ( , ) = 11 = 1 ( 11 − ( 22 + 33 )) (1) where the stresses depend on the applied load . Thus, it is required to find such a value of the parameter vector = ( , ) that would deliver the minimum of the function: = ∑ ( ( ) − ( , )) 2 =1 (2) where is the deformation value obtained from FBGs under load , ( , ) is the calculated strain values obtained by simulation at the load . In the case of several control points, the multiobjective optimization problem is transformed into a single-line optimization problem using the weighting coefficients . Then the error estimation function is written as a sum: = ∑ ̇ =1 (3) where is the number of control points, is the functional (2) for the control point , the weighting factors ̇ are chosen based on the parameters and initial conditions of the specific task (for example, they can be related to the location of control points). The initial value of the parameter vector corresponds to known values of the layer constants. To estimate the increment of the parameter vector Δ according to the Levenberg-Marquardt algorithm, the following formula is used: Δ = ( + diag[ ]) −1 ( − ) (4) where and are vectors of strain corresponded to load values = 1 , 2 , 3,… obtained from FBG sensors and FE model, respectively; parameter controls approximation speed, is a Jacobian of ( , ) function with the current parameters values :