PSI - Issue 84

Gianluca Bruno et al. / Procedia Structural Integrity 84 (2026) 240–247 = [ (1), … , (ℎ), … , ( )] After, Singular Value Decomposition (SVD) is applied to the matrix : = (5) where ∈ℝ × is an orthogonal matrix whose columns ( is the column i of matrix , termed as POD mode), named left singular vectors, form a basis for the solution space; Σ∈ℝ × is a diagonal matrix containing the singular values 1 ≥ 2 ≥⋯≥0 in descending order; V∈ℝ × is an orthogonal matrix whose columns are the right singular vectors. The reduced basis Φ is defined by selecting the first left singular vectors (columns of ) corresponding to the most energetic singular values. = [ 1 ,…, …, ] (6) This truncation operation ensures that the generated subspace captures the maximum possible variance of the original data, minimizing the projection error. 3. Proposed framework The proposed framework is articulated in six consequential steps. The first step consists of defining experimental target quantities, , which are representative of the dynamic behavior of the structure. For the case at hand, stand for natural frequencies of the structure, obtained by performing OMA over data from a monitoring campaign. In the second phase, the FOM of the structure is defined, which represents the numerical reference for two fundamental aspects: (a) the generation of the dataset, , necessary for the creation of the ROM; (b) the full representation of the structural behaviour at the end of the process. The FOM should include a detailed definition of the structural geometry, the mechanical properties, the constraint conditions. In this phase, the sources of uncertainty are defined and are indicated through the vector . The dataset is built by sampling from the parameter space within physically plausible intervals. For each combination (ℎ) , with ℎ = 1,…, , the FOM is analysed to obtain the modal quantities of interest and the corresponding mass and stiffness matrices. In the third phase, the dataset is used to define the ROM, which constitutes the numerical environment on which the DRL agent operates and from which the simulated target variables are obtained. To create ROM, the snapshot matrix is assembled and SVD technique is applied, in order to extract the first left singular vectors, corresponding to the most energetic modal directions. These vectors define the reduced basis Φ , which allows to project mass and stiffness matrices of the FOM into a lower-dimensional space. Once the reduction basis is obtained, the ROM results explicitly dependent on the uncertain parameters , which are the quantities to update by the agent during the learning process. To update the matrices and , a Taylor first-order expansion is used, as suggested by Khodaparast et al. (2008). As for example, the stiffness matrix is updated as follows ( (ℎ)) ≈ ( (0))+∑ (ℎ) | (0) ( = 1 (ℎ)− (0)) (7) where ( (ℎ)) is the stiffness matrix evaluated for the combination of uncertain parameters (ℎ) related to the scenario ℎ ; ( (0)) is the stiffness matrix for the reference values of the parameters (0)= [ 1 (0), … , (0), … , (0)] ; (ℎ) is the sensitivity matrix, i.e. the partial derivative of the stiffness matrix with respect to the parameter (ℎ) in the point of expansion; ( (ℎ)− (0)) is the parametric variation of the uncertain parameter with respect to the reference. In this way, each modification proposed by the agent is reflected in a consistent update of the matrices and the corresponding modal quantities to match with . The updating process is developed in the fourth phase, in which the ROM is integrated into a DRL environment. The state of the environment ( ) is defined as a combination of the information available at time , including the modal quantities calculated by ( ) and the current values of the uncertain parameters ( ) . For each timestep , (4) 243