The objective of this thesis is to provide a mathematical and computational framework for the proactive maintenance of complex systems with a particular application to structural health monitoring (SHM). SHM techniques rely primarily on sensor responses to assess the risk associated with a structure of interest and seek to provide a support for automated decision-making strategy. An efficient integration of experimental measurements and numerical models is needed to exhaustively describe environmental and operational scenarios that a structure undergoes during its life time.

We propose a simulation-based approach that combines the solution of a parametric time-dependent partial differential equation (PDE) for multiple input parameters with data-driven techniques to discriminate between healthy and damaged configurations. This process exploits an offline-online decomposition of tasks. The dataset of synthetic sensor measurements is generated offline by repeatedly solving a parametric PDE for a predefined configuration under several healthy variations. A reduced order model is employed to overcome the computational bottleneck associated with the many-query context by building a projection-based reduced basis method in combination with a Laplace-domain solver. Weeks method is adopted to numerically invert the Laplace transform and transform the signals to the time domain. The datasets are used to train various one-class machine learning algorithms in a semi-supervised setting, sensor by sensor. Finally, the outputs of the classifiers are used to assess the state of damage of the structure online. Using a decision-level fusion strategy, we provide insight on the existence, location, and severity of possible damages.

The performance of SHM depends critically on the quality of the sensor measurements although their availability is often limited due to budget constraints and installation difficulties. We therefore propose a strategy to systematically place a fixed number of sensors on a structure of interest to minimize uncertainty at unsensed locations. The so-called inducing points, an outcome of sparse Gaussian processes, originally introduced to overcome the computational burden associated with performing a regression task with standard Gaussian processes, are here used to guide the sensor placement. A clustering approach is employed to select the sensor locations among a set of inducing points, computed for different input parameters.

We apply this methodology to 2D and 3D problems to mimic the vibrational behavior of complex structures under the effect of an active source and show the effectiveness of the approach for (i) detecting damaged geometries and (ii) identifying the locations for a network of sensors. This framework considers the realistic case where damage types and locations are a priori unknown, thus overcoming the main limitation of existing simulation-based damage detection and sensor placement strategies for SHM.