Analysis of Stationary Random Responses for Non-Parametric Probabilistic Systems

Uncertainties in finite element (FE) model updating arise from two main sources: measurement noise and modeling errors. The latter includes model parameter uncertainty and model uncertainty itself. Among these sources of uncertainty, model uncertainty has been proven to be the most influential source of error in FE model updating, which is particularly important when using the updated model for damage identification (DID) purposes. This paper investigates the effects of model uncertainty when updating mechanics-based nonlinear FE models of building structures subjected to earthquake excitation. To solve the parameter estimation problem, the unscented Kalman filter is used as parameter estimation tool. Numerically simulated response data of two state-of-the-art nonlinear FE models of building structures designed according to modern design codes are used as application examples. A two-dimensional steel building and a three-dimensional reinforced concrete building, both subjected to seismic base excitation, are analyzed for different types and levels of model uncertainty. The results show that model uncertainty may have significant detrimental effects when using the updated FE model for DID, chiefly in the case of large modeling uncertainty. Although a good match between the measured (observed) and FE predicted responses is usually achieved, unobserved responses at global and local levels often show significant errors.

Comparative study of damped FE model updating methods

Model updating-based dynamic collapse analysis of a RC cable-stayed bridge under earthquakes

Single cell RNA-seq (scRNAseq) workflows typically start with a count matrix and end with the clustering of sampled cells. While a range of methods have been developed to cluster scRNAseq datasets, no theoretical tools exist to explain why a particular cluster exists or why a hypothesized cluster is missing. Recently, several authors have shown that eigenvalues of scRNAseq count matrices can be approximated using random matrix models. In this work, we extend these previous works to the study of a scRNAseq workflow. We model scaled count matrices using random matrices with normally distributed entries. Using these random matrix models, we quantify the differential expression of a cluster and develop predictions for the workflow, and in particular clustering, as a function of the differential expression. We also use results from random matrix theory (RMT) to develop predictive formulas for portions of the scRNAseq workflow. Using simulated and real datasets, we show that our predictions are accurate if certain conditions hold on differential expression, with our RMT based predictions requiring particularly stringent condition. We find that real datasets violate these conditions, leading to bias in our predictions, but our predictions are better than a naive estimator and we point out future work that can improve the predictions. To our knowledge, our formulas represents the first predictive results for scRNAseq workflows.

Identification, Model Updating, and Validation of a Steel Twin Deck Curved Cable‐Stayed Footbridge

The usual model updating method may be considered to be deterministic since it uses measurements from a single test system to correct a nominal finite element model. There may however be variability in seemingly identical test structures and uncertainties in the finite element model. Variability in test structures may arise from many sources including geometric tolerances and the manufacturing process, and modelling uncertainties may result from the use of nominal material properties, ill-defined joint stiffnesses and rigid boundary conditions. In this paper the theory of stochastic model updating using a Monte-Carlo inverse procedure with multiple sets of experimental results is explained and then applied to the case of a simulated three degree-of-freedom system, which is used to fix ideas and also to illustrate some of the practical limitations of the method. In the companion paper, stochastic model updating is applied to a benchmark structure using a contact finite element model that includes common uncertainties in the modelling of the spot welds.

Rupture prediction of abdominal aortic aneurysms (AAAs) remains a clinical challenge. Finite element analysis (FEA) may allow for improved identification for intervention timing, but the method needs further substantiation. In this study, experimental photoelastic method and finite element techniques were compared using an idealised AAA geometry. There was good agreement between the numerical and experimental results. At the proximal and distal end of the AAA model, the maximum differences in principle strain for an internal pressure of 120 mmHg had differences ranging from 0.03 to 10.01%. The maximum difference in principle strain for the photoelastic and the finite element model at a pressure of 120 mmHg was 0.167 and 0.158, respectively. The current research strengthens the case for using FEA as an adjunct to the current clinical practice of utilising diameter measurement for intervention timing.

Motivated by the asymptotic collective behavior of random and deterministic matrices, we propose an approximation (called "free deterministic equivalent") to quite general random matrix models, by replacing the matrices with operators satisfying certain freeness relations. We comment on the relation between our free deterministic equivalent and deterministic equivalents considered in the engineering literature. We do not only consider the case of square matrices, but also show how rectangular matrices can be treated. Furthermore, we emphasize how operator-valued free probability techniques can be used to solve our free deterministic equivalents. As an illustration of our methods we show how the free deterministic equivalent of a random matrix model from [6] can be treated and we thus recover in a conceptual way the results from [6]. On a technical level, we generalize a result from scalar-valued free probability, by showing that randomly rotated deterministic matrices of different sizes are asymptotically free from deterministic rectangular matrices, with amalgamation over a certain algebra of projections. In Appendix A, we show how estimates for differences between Cauchy transforms can be extended from a neighborhood of infinity to a region close to the real axis. This is of some relevance if one wants to compare the original random matrix problem with its free deterministic equivalent.

In various applications it is important to determine dynamically equivalent spatial finite element (FE) model of complex structures. For instance, obtaining the FE model of an existing aerospace structure is a major requirement for reliable aeroelastic analysis. In such applications a reliable FE model may not be always available, and when this is so a dynamically equivalent FE model derived from modal test will be very useful. This paper presents a noble method to determine spatial FE model of a structure by using experimentally measured modal data along with the connectivity information of measurement points. The method is based on the mass and stiffness orthogonality equations written using experimentally determined mode shapes and natural frequencies. These equations are solved for geometric and material properties constituting global spatial mass and stiffness matrices of an initial FE model. Starting from this initial FE model, mass and stiffness orthogonality equations are updated iteratively employing experimentally obtained natural frequencies and corresponding eigenvectors from the FE model. Iterations are continued until eigensolution of the updated FE model closely correlates with experimentally measured modal data. A simulated case study on GARTEUR scaled aircraft model is presented in order to demonstrate the applicability of the method.

Three‐dimensional (3D) finite element (FE) modelling of axial flux permanent magnet (AFPM) motors is one of the time‐consuming tasks which must be completed before manufacturing the prototype motor. A simple, fast, accurate, and effective numerical modelling approach for such 3D problems would be particularly useful for researchers. This study presents simple and effective 2D FE modelling approaches for AFPM motors with good accuracy with less computational effort. The modelling approaches are based on a number of 2D models or planes with inner rotor, outer rotor, or linear motor topologies. All the 2D planes with either topology are superposed together to represent the real AFPM machine. These approaches are examined in detail with various types of experimentally verified AFPM motors. Results obtained from the proposed modelling approaches are verified with 3D FE analyses and experimental data. The results indicate that the proposed 2D modelling approaches can significantly reduce the computation time with good accuracy as opposed to conventional 3D FE modelling of AFPM motors.

This work presents a method for uncertainty propagation that is consistent with the “building-block approach” in which components of a system are tested and validated individually instead of an integrated vehicle test and validation being performed. The approach gives a unified methodology for representing and quantifying uncertainty in a Hurty/Craig-Bampton component based on component test results and propagating the uncertainty in component models into system-level predictions. Uncertainty in the Hurty/Craig-Bampton representations is quantified using a new hybrid parametric variation approach based on Soize’s maximum entropy method. The proposed approach combines parametric and nonparametric uncertainty by treating the Hurty/Craig-Bampton fixed-interface eigenvalues as random variables and treating the corresponding mass and stiffness as random matrices. The proposed method offers several advantages over traditional approaches to uncertainty quantification in structural dynamics: the number of parametric random variables is relatively small compared to the usually large number of potential random finite element model parameters; therefore, time-consuming parametric sensitivity studies do not have to be performed. In addition, nonparametric model-form uncertainty is easily included using random matrix theory. The method requires the selection of dispersion values for the Hurty/Craig-Bampton fixed-interface eigenvalues and the corresponding mass and stiffness matrices. Test/analysis frequency error is used to identify the fixed-interface eigenvalue dispersions, and test/analysis cross-orthogonality is used to identify the Hurty/Craig-Bampton stiffness matrix dispersion value. Currently, the mass matrix dispersion is based on engineering judgment, past experience, and historical results. The proposed uncertainty quantification methodology is applied to the Space Launch System liftoff configuration. Robustness of the attitude control system is studied by propagating derived component uncertainty models into gain uncertainty in specific transfer functions relating engine inputs to rate sensors on the core stage, and frequency uncertainty for the fundamental bending and roll modes.

Uncertainty propagation in FE modelling of a fire resistance test using fractional factorial design based model reduction and deterministic sampling

Seismic response analysis of structural system subjected to multiple support excitation

Mixed-dimensional modeling of structures with thin coatings using rotation-free elements

Direct probability integral method for stochastic response analysis of static and dynamic structural systems