Igor Kavrakov

Gledson Rodrigo Tondo, Sebastian Rau, Igor Kavrakov et al.

Machine learning models trained with structural health monitoring data have become a powerful tool for system identification. This paper presents a physics-informed Gaussian process (GP) model for Timoshenko beam elements. The model is constructed as a multi-output GP with covariance and cross-covariance kernels analytically derived based on the differential equations for deflections, rotations, strains, bending moments, shear forces and applied loads. Stiffness identification is performed in a Bayesian format by maximising a posterior model through a Markov chain Monte Carlo method, yielding a stochastic model for the structural parameters. The optimised GP model is further employed for probabilistic predictions of unobserved responses. Additionally, an entropy-based method for physics-informed sensor placement optimisation is presented, exploiting heterogeneous sensor position information and structural boundary conditions built into the GP model. Results demonstrate that the proposed approach is effective at identifying structural parameters and is capable of fusing data from heterogeneous and multi-fidelity sensors. Probabilistic predictions of structural responses and internal forces are in closer agreement with measured data. We validate our model with an experimental setup and discuss the quality and uncertainty of the obtained results. The proposed approach has potential applications in the field of structural health monitoring (SHM) for both mechanical and structural systems.

Gledson Rodrigo Tondo, Igor Kavrakov, Guido Morgenthal

Long-span bridges are subjected to a multitude of dynamic excitations during their lifespan. To account for their effects on the structural system, several load models are used during design to simulate the conditions the structure is likely to experience. These models are based on different simplifying assumptions and are generally guided by parameters that are stochastically identified from measurement data, making their outputs inherently uncertain. This paper presents a probabilistic physics-informed machine-learning framework based on Gaussian process regression for reconstructing dynamic forces based on measured deflections, velocities, or accelerations. The model can work with incomplete and contaminated data and offers a natural regularization approach to account for noise in the measurement system. An application of the developed framework is given by an aerodynamic analysis of the Great Belt East Bridge. The aerodynamic response is calculated numerically based on the quasi-steady model, and the underlying forces are reconstructed using sparse and noisy measurements. Results indicate a good agreement between the applied and the predicted dynamic load and can be extended to calculate global responses and the resulting internal forces. Uses of the developed framework include validation of design models and assumptions, as well as prognosis of responses to assist in damage detection and structural health monitoring.

Gledson Rodrigo Tondo, Sebastian Rau, Igor Kavrakov et al.

A physics-informed machine learning model, in the form of a multi-output Gaussian process, is formulated using the Euler-Bernoulli beam equation. Given appropriate datasets, the model can be used to regress the analytical value of the structure's bending stiffness, interpolate responses, and make probabilistic inferences on latent physical quantities. The developed model is applied on a numerically simulated cantilever beam, where the regressed bending stiffness is evaluated and the influence measurement noise on the prediction quality is investigated. Further, the regressed probabilistic stiffness distribution is used in a structural health monitoring context, where the Mahalanobis distance is employed to reason about the possible location and extent of damage in the structural system. To validate the developed framework, an experiment is conducted and measured heterogeneous datasets are used to update the assumed analytical structural model.

Igor Kavrakov, Yaswanth Sai Jetti, Ahmet Oguzhan Yuksel et al.

We present an approach for synthesising observational data with elastodynamic finite element models by extending the statistical finite element method (statFEM) framework. The proposed formulation adopts a Bayesian filtering approach to account for uncertainties in the data, the finite element model, and the discrepancies between the model and the physical system. Observational data are assimilated while the state of the spatially discretised finite element problem is advanced in time using the stochastic variant of the explicit Newmark scheme. The prior probability density of the state is obtained by solving an incremental probabilistic forward problem, and the corresponding posterior density is obtained by conditioning on the data available at each time step. In the probabilistic forward problem, spatio-temporal Gaussian random fields representing the forcing, model misspecification, and material parameters are specified via their stochastic PDE formulation. The resulting non-Gaussian prior probability density is approximated using a perturbation approach, yielding a Gaussian posterior with closed-form mean and covariance. The hyperparameters of the random field representing model misspecification are calibrated by maximising the marginal likelihood of the data. The proposed approach is illustrated on one- and two-dimensional elastodynamic examples with synthetic data.

Gledson Rodrigo Tondo, Igor Kavrakov, Guido Morgenthal

Accurate modeling of aerodynamic loads is essential for understanding and predicting the responses of complex structural systems. However, these models often rely on simplifications of the true physical forces, introducing assumptions that can limit their accuracy. Validating such models becomes particularly challenging in the presence of noisy or incomplete data. To address this, we introduce a probabilistic physics-informed machine learning approach designed to reconstruct the underlying aerodynamic loads from noisy measurements of structural dynamic responses. The model avoids overfitting, eliminates the need for regularization schemes, and allows for the use of heterogeneous and multi-fidelity data during the training process. The efficacy of the approach is demonstrated through the reconstruction of aerodynamic loads on the Great Belt East Bridge, simulated under a linear unsteady assumption. Results show a strong agreement between true and predicted loads, particularly related to root mean squared errors, magnitude, phase angle and peak values of the signals. The method for load reconstructing holds broad applicability, such as modeling validation, future load estimation, and structural damage prognosis.

Gledson Rodrigo Tondo, Igor Kavrakov, Guido Morgenthal

Knowledge of the force time history of a structure is essential to assess its behaviour, ensure safety and maintain reliability. However, direct measurement of external forces is often challenging due to sensor limitations, unknown force characteristics, or inaccessible load points. This paper presents an efficient dynamic load reconstruction method using physics-informed Gaussian processes (GP) based on frequency-sparse Fourier basis functions. The GP's covariance matrices are built using the description of the system dynamics, and the model is trained using structural response measurements. This provides support and interpretability to the machine learning model, in contrast to purely data-driven methods. In addition, the model filters out irrelevant components in the Fourier basis function by leveraging the sparsity of structural responses in the frequency domain, thereby reducing computational complexity during optimization. The trained model for structural responses is then integrated with the differential equation for a harmonic oscillator, creating a probabilistic dynamic load model that predicts load patterns without requiring force data during training. The model's effectiveness is validated through two case studies: a numerical model of a wind-excited 76-story building and an experiment using a physical scale model of the Lillebælt Bridge in Denmark, excited by a servo motor. For both cases, validation of the reconstructed forces is provided using comparison metrics for several signal properties. The developed model holds potential for applications in structural health monitoring, damage prognosis, and load model validation.

Igor Kavrakov, Gledson Rodrigo Tondo, Guido Morgenthal

Advancements in machine learning and an abundance of structural monitoring data have inspired the integration of mechanical models with probabilistic models to identify a structure's state and quantify the uncertainty of its physical parameters and response. In this paper, we propose an inference methodology for classical Kirchhoff-Love plates via physics-informed Gaussian Processes (GP). A probabilistic model is formulated as a multi-output GP by placing a GP prior on the deflection and deriving the covariance function using the linear differential operators of the plate governing equations. The posteriors of the flexural rigidity, hyperparameters, and plate response are inferred in a Bayesian manner using Markov chain Monte Carlo (MCMC) sampling from noisy measurements. We demonstrate the applicability with two examples: a simply supported plate subjected to a sinusoidal load and a fixed plate subjected to a uniform load. The results illustrate how the proposed methodology can be employed to perform stochastic inference for plate rigidity and physical quantities by integrating measurements from various sensor types and qualities. Potential applications of the presented methodology are in structural health monitoring and uncertainty quantification of plate-like structures.

Igor Kavrakov, Allan McRobie, Guido Morgenthal

An abundant amount of data gathered during wind tunnel testing and health monitoring of structures inspires the use of machine learning methods to replicate the wind forces. This paper presents a data-driven Gaussian Process-Nonlinear Finite Impulse Response (GP-NFIR) model of the nonlinear self-excited forces acting on structures. Constructed in a nondimensional form, the model takes the effective wind angle of attack as lagged exogenous input and outputs a probability distribution of the forces. The nonlinear input/output function is modeled by a GP regression. Consequently, the model is nonparametric, thereby circumventing to set up the function's structure a priori. The training input is designed as random harmonic motion consisting of vertical and rotational displacements. Once trained, the model can predict the aerodynamic forces for both prescribed input motion and aeroelastic analysis. The concept is first verified for a flat plate's analytical solution by predicting the self-excited forces and flutter velocity. Finally, the framework is applied to a streamlined and bluff bridge deck based on Computational Fluid Dynamics (CFD) data. The model's ability to predict nonlinear aerodynamic forces, flutter velocity, and post-flutter behavior are highlighted. Applications of the framework are foreseen in the structural analysis during the design and monitoring of slender line-like structures.