MS20.6 - UQ and Probabilistic Learning in Computational Dynamics

Structural health monitoring (SHM), relies on evaluating how certain features change over time (such as natural frequencies), to detect changes indicative of damage. However, damage-sensitive features often respond to benign variations, such as changes in temperature, in addition to damage. These normal variations can mimic or mask damage, and can therefore affect the efficacy and generalisation of SHM technologies. Population-based SHM (PBSHM) aims to share valuable information, including normal condition and damage states, among similar structures. The aim of the current work is to investigate the effects of temperature variation and identify methods to account for these changes, using PBSHM principles. A hierarchical Bayesian approach was used to develop a combined probabilistic frequency response function (FRF) model using FRFs from a full-scale composite helicopter blade. The FRFs were computed using vibration data collected with the blade exposed to various temperatures in an environmental chamber. Functions representing how the natural frequency and modal damping changed according to temperature were determined, and the results were extrapolated to temperatures not used in model training.

Printed Circuit Boards (PCB) used in electric vehicles are subject to mechanical vibrations that affect reliability. Here, especially solder joints are a critical point of PCBs that can be damaged by mechanical vibrations. The surface strain at the solder location is often seen as a critical mechanical parameter for the failure of solder joints and is used as a design criterion. Therefore, simulation during the design process needs to provide accurate results for the surface strains on the PCBs under specified loads. Due to model parameter uncertainty, a parameter calibration and model validation step is usually necessary. In this work, Bayesian model updating using experimental measurements is performed on PCBs. A comparison of the weights of the investigated PCBs shows that the material parameters are subject to considerable uncertainty. These uncertainties are likely due to small variations in the material composition or the production process. Thus, hierarchical Bayesian model updating is chosen as model updating method. This approach makes it possible to determine probability distributions that reflect the uncertainty of the model parameters. Since the PCBs only weigh a few grams, a sensor and its electric cables could have an impact on their mechanical behavior. Therefore, the dynamic behavior is measured by a laser Doppler vibrometer (LDV). The measurements are carried out on printed circuit boards that are fixed with adhesive joints. The model updating process considers modal data as well as frequency response functions. The updated parameters include the stiffness and damping properties of the PCB’s basic material and the adhesive. Since the updated numerical model of the PCB is to be used for the simulation of surface strains at the solder joints, further measurements are necessary for model validation. For this purpose, additional measurements of surface strains at different excitation frequencies are carried out with strain gauges.

Aging vibration-prone structures may require up-to-date proofs of their load-bearing capacity for retrofitting or life-cycle assessments. The structural vibration response is affected by naturally variable structural properties and loads, known as aleatory uncertainties. With modeling, additional epistemic uncertainties are introduced due to missing knowledge of model parameters. The model responses then include both types of uncertainties in a mixed and nested form. Quantifying these polymorphic uncertainties is particularly important in structural dynamics, where a deterministic model may significantly underestimate responses, such as resonance phenomena.
A case study was conducted to characterize the vibration behavior of a guyed mast under polymorphic uncertainties. A simplified linear structural model was built using commercially available FEM software. Polymorphic uncertainties in the input parameters were modeled and propagated using in-house code.
Aleatory uncertainties were modeled by probability theory, and a quasi-Monte Carlo simulation based on importance sampling facilitates an efficient reuse of computed samples. Epistemic uncertainties were modeled by evidence / belief function theory and their propagation was performed using approximate interval optimizations, avoiding surrogate models to avoid non-physical results. Particular emphasis was placed on computational efficiency, as the developed method is designed to be used for more complex numerical models. Statistics and aggregation of belief functions were used in conjunction with sensitivity analyses to yield explainable results.
The case study scenario is a retrofit of a guyed mast based on outdated design documents. Epistemically uncertain parameters include unknown structural damping, cross-section tolerances and material properties, the additional mass from antennas and cables, and pre-tensioning forces of the guy wires. Aleatory uncertainties arise from temperature effects on the guy wires and the viscosity of the dampers, as well as from icing in winter. The results highlight the natural variations of resonant frequencies, the characterization of resonant bands due to epistemic uncertainties, and the effectiveness of a deterministically designed Tuned Mass Damper in a structural model influenced by polymorphic uncertainties.
The developed methodology is applicable to moderately computationally expensive models of any type and does not require building surrogate models. Model runs are efficiently re-used and the uncertainty propagation can be highly parallelized. At the same computational cost, a pure stochastic treatment of uncertainties achieves slightly better variances, while concealing the influence of missing knowledge. The method reveals where enhancing knowledge about partly-known model parameters is beneficial and permits robust retrofit designs or increased confidence in life-cycle estimates.

This paper presents a methodology [1] devoted to updating the control parameters of large computational models in linear structural dynamics. In particular, we are interested in the linear vibrations in the frequency domain of complex structures that are characterized by the presence of numerous structural elastic modes, in particular local modes. These local modes arise from the multilevel nature of the structure and are spread throughout the entire frequency band. In such a context, it is assumed that one call to the computational model entails a very high computational cost, such that no more than a hundred calls can reasonably be considered. Furthermore, it is assumed that only one target response is available for identifying the control parameters. The optimization problem is not convex and since only a small number of realizations is affordable, it is proposed to use the Probabilistic Learning on Manifold (PLoM) [2] to generate additional realizations under constraints. Then, the cost function is evaluated using conditional statistics. The additional realizations are learned based on the few (less than a hundred) realizations that constitute the training set. The PloM method is particularly tailored to the case of small data and consequently, it is well indicated in this case. As a numerical application, the proposed methodology using PLoM and conditional statistics is validated for the case of a simple structure that features high modal density. [1] O. Ezvan, C. Soize, C. Desceliers, R. Ghanem, Updating an uncertain and expensive computational model in structural dynamics based on one single target FRF using a probabilistic learning tool, submitted (2022). [2] C. Soize, R. Ghanem, Probabilistic learning on manifolds (PLoM) with partition, International Journal for Numerical Methods in Engineering, doi: 10.1002/nme.6856, 123(1), 268-290 (2022).