PSI - Issue 81

Andrejs Kovalovs et al. / Procedia Structural Integrity 81 (2026) 388–395

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where b 0 , b k , b kk , b km are regression coefficients determined by least-squares fitting from the 101 FE data points. Although the polynomial model is relatively simple, previous studies (Rikards, 2003; Rikards and Auzins, 2004) have shown that a quadratic approximation adequately captures the smooth dependence of eigenfrequencies on material stiffness parameters for laminated composites. This polynomial, though simple, captures the smooth variation of frequencies with elastic constants remarkably well. To verify its adequacy, the coefficients of determination R 2 were computed. For all considered cases, the fits achieved R 2 > 0, indicating excellent correlation between the approximated and computed frequencies. The residuals exhibited random distribution without systematic bias, confirming that higher-order terms were unnecessary. Analysis of the approximation equations showed that, for both unidirectional and twill-weave laminates, the longitudinal modulus E 1 and the in-plane shear modulus G 12 exert the strongest influence on the modal frequencies, particularly for the first six modes. The transverse modulus E 2 and Poisson’s ratio ν 12 were also found to be influential but to a lesser extent. In contrast, the out-of-plane constants E 3 and G 23 had little effect on the frequency spectrum of thin plates, confirming that they can be fixed at nominal values without loss of accuracy. This dimensionality reduction – from six to four active variables – significantly improves the conditioning of the optimization problem and reduces the computational effort. The constructed response surfaces were visualized to confirm the smoothness and monotonic trends of frequency variation with respect to key parameters. Fig. 2 (a-c) illustrates a representative example of the response surface for the bending mode (1,1) of the unidirectional plate, expressed as a function of E 1 and G 12 . For the twill-weave plate, the response surface (Fig. 2 d-f) exhibits a nearly parabolic contour shape, confirming weak nonlinearity and validating the use of a quadratic model for interpolation.

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Fig. 2. Example of a response surface for the fundamental bending mode (1,1) frequency for unidirectional (a – c) and twill-weave (d – f) CFRP laminates.

The derived polynomial models were subsequently used as response surface functions in the inverse identification procedure. By substituting the response surfaces into the objective functional Eq. (2), the minimization problem was solved efficiently using gradient-free optimization, while retaining the predictive accuracy of the full FE model. A hybrid global-local optimization strategy was applied: a global Differential Evolution search generated candidate solutions across the entire domain, followed by local refinement using the Nelder-Mead simplex algorithm. Alternative algorithms – Dual Annealing, CMA-ES, and Bayesian optimization – were also tested and yielded consistent results. Convergence to the optimum was typically achieved within 40-50 iterations. The average computational time for a complete identification run was below one minute on a standard workstation. Validation tests with synthetic “experimental” frequencies containing ±0.2% random noise confirmed recovery of the original para meters within 1% deviation, demonstrating the robustness and efficiency of the developed RSM-based optimization procedure.