Information exists in various forms and a better utilization of the available information can benefit the system awareness and response predictions. The focus of this dissertation is on the fusion of different types of information using Bayesian-Entropy method. The Maximum Entropy method in information theory introduces a unique way of handling information in the form of constraints. The Bayesian-Entropy (BE) principle is proposed to integrate the Bayes’ theorem and Maximum Entropy method to encode extra information. The posterior distribution in Bayesian-Entropy method has a Bayesian part to handle point observation data, and an Entropy part that encodes constraints, such as statistical moment information, range information and general function between variables. The proposed method is then extended to its network format as Bayesian Entropy Network (BEN), which serves as a generalized information fusion tool for diagnostics, prognostics, and surrogate modeling.

The proposed BEN is demonstrated and validated with extensive engineering applications. The BEN method is first demonstrated for diagnostics of gas pipelines and metal/composite plates for damage diagnostics. Both empirical knowledge and physics model are integrated with direct observations to improve the accuracy for diagnostics and to reduce the training samples. Next, the BEN is demonstrated in prognostics and safety assessment in air traffic management system. Various information types, such as human concepts, variable correlation functions, physical constraints, and tendency data, are fused in BEN to enhance the safety assessment and risk prediction in the National Airspace System (NAS). Following this, the BE principle is applied in surrogate modeling. Multiple algorithms are proposed based on different type of information encoding, such as Bayesian-Entropy Linear Regression (BELR), Bayesian-Entropy Semiparametric Gaussian Process (BESGP), and Bayesian-Entropy Gaussian Process (BEGP) are demonstrated with numerical toy problems and practical engineering analysis. The results show that the major benefits are the superior prediction/extrapolation performance and significant reduction of training samples by using additional physics/knowledge as constraints. The proposed BEN offers a systematic and rigorous way to incorporate various information sources. Several major conclusions are drawn based on the proposed study.

The focus of this dissertation is first on understanding the difficulties involved in constructing reduced order models of structures that exhibit a strong nonlinearity/strongly nonlinear events such as snap-through, buckling (local or global), mode switching, symmetry breaking. Next, based on this understanding, it is desired to modify/extend the current Nonlinear Reduced Order Modeling (NLROM) methodology, basis selection and/or identification methodology, to obtain reliable reduced order models of these structures. Focusing on these goals, the work carried out addressed more specifically the following issues:

i) optimization of the basis to capture at best the response in the smallest number of modes,

ii) improved identification of the reduced order model stiffness coefficients,

iii) detection of strongly nonlinear events using NLROM.

For the first issue, an approach was proposed to rotate a limited number of linear modes to become more dominant in the response of the structure. This step was achieved through a proper orthogonal decomposition of the projection on these linear modes of a series of representative nonlinear displacements. This rotation does not expand the modal space but renders that part of the basis more efficient, the identification of stiffness coefficients more reliable, and the selection of dual modes more compact. In fact, a separate approach was also proposed for an independent optimization of the duals. Regarding the second issue, two tuning approaches of the stiffness coefficients were proposed to improve the identification of a limited set of critical coefficients based on independent response data of the structure. Both approaches led to a significant improvement of the static prediction for the clamped-clamped curved beam model. Extensive validations of the NLROMs based on the above novel approaches was carried out by comparisons with full finite element response data. The third issue, the detection of nonlinear events, was finally addressed by building connections between the eigenvalues of the finite element software (Nastran here) and NLROM tangent stiffness matrices and the occurrence of the ‘events’ which is further extended to the assessment of the accuracy with which the NLROM captures the full finite element behavior after the event has occurred.

The problem of modeling and controlling the distribution of a multi-agent system has recently evolved into an interdisciplinary effort. When the agent population is very large, i.e., at least on the order of hundreds of agents, it is important that techniques for analyzing and controlling the system scale well with the number of agents. One scalable approach to characterizing the behavior of a multi-agent system is possible when the agents' states evolve over time according to a Markov process. In this case, the density of agents over space and time is governed by a set of difference or differential equations known as a {\it mean-field model}, whose parameters determine the stochastic control policies of the individual agents. These models often have the advantage of being easier to analyze than the individual agent dynamics. Mean-field models have been used to describe the behavior of chemical reaction networks, biological collectives such as social insect colonies, and more recently, swarms of robots that, like natural swarms, consist of hundreds or thousands of agents that are individually limited in capability but can coordinate to achieve a particular collective goal.

This dissertation presents a control-theoretic analysis of mean-field models for which the agent dynamics are governed by either a continuous-time Markov chain on an arbitrary state space, or a discrete-time Markov chain on a continuous state space. Three main problems are investigated. First, the problem of stabilization is addressed, that is, the design of transition probabilities/rates of the Markov process (the agent control parameters) that make a target distribution, satisfying certain conditions, invariant. Such a control approach could be used to achieve desired multi-agent distributions for spatial coverage and task allocation. However, the convergence of the multi-agent distribution to the designed equilibrium does not imply the convergence of the individual agents to fixed states. To prevent the agents from continuing to transition between states once the target distribution is reached, and thus potentially waste energy, the second problem addressed within this dissertation is the construction of feedback control laws that prevent agents from transitioning once the equilibrium distribution is reached. The third problem addressed is the computation of optimized transition probabilities/rates that maximize the speed at which the system converges to the target distribution.

Shock loading produces a compressive stress pulse with steep gradients in density, temperature, and pressure that are also often modeled as discontinuities. When a material is subject to these dynamic (shock) loading conditions, fracture and deformation patterns due to spall damage can arise. Spallation is a dynamic material failure that is caused by the nucleation, growth, and coalescence of voids, with possible ejection of the surface of the material. Intrinsic defects, such as grain boundaries are the preferred initiation sites of spall damage in high purity materials. The focus of this research is to study the phenomena that cause void nucleation and growth at a particular grain boundary (GB), chosen to maximize spall damage localization.

Bicrystal samples were shock loaded using flyer-plates via light gas gun and direct laser ablation. Stress, pulse duration, and crystal orientation along the shock direction were varied for a fixed boundary misorientation to determine thresholds for void nucleation and coalescence as functions of these parameters. Pressures for gas gun experiments ranged from 2 to 5 GPa, while pressures for laser ablation experiments varied from 17 to 25 GPa. Samples were soft recovered to perform damage characterization using electron backscattering diffraction (EBSD) and Scanning Electron Microscopy (SEM). Results showed a 14% difference in the thresholds for void nucleation and coalescence between samples with different orientations along the shock direction, which were affected by pulse duration and stress level. Fractography on boundaries with strong damage localization showed many small voids, indicating they experience rapid nucleation, causing early coalescence. Composition analysis was also performed to determine the effect of impurities on damage evolution. Results showed that higher levels of impurities led to more damage. ABAQUS/Explicit models were developed to simulate flyer-plate impact and void growth with the same crystal orientations and experimental conditions. Results are able to match the damage seen in each grain of the target experimentally. The Taylor Factor mismatch at the boundary can also be observed in the model with the higher Taylor Factor grain exhibiting more damage.

Numerous works have addressed the control of multi-robot systems for coverage, mapping, navigation, and task allocation problems. In addition to classical microscopic approaches to multi-robot problems, which model the actions and decisions of individual robots, lately, there has been a focus on macroscopic or Eulerian approaches. In these approaches, the population of robots is represented as a continuum that evolves according to a mean-field model, which is directly designed such that the corresponding robot control policies produce target collective behaviours.



This dissertation presents a control-theoretic analysis of three types of mean-field models proposed in the literature for modelling and control of large-scale multi-agent systems, including robotic swarms. These mean-field models are Kolmogorov forward equations of stochastic processes, and their analysis is motivated by the fact that as the number of agents tends to infinity, the empirical measure associated with the agents converges to the solution of these models. Hence, the problem of transporting a swarm of agents from one distribution to another can be posed as a control problem for the forward equation of the process that determines the time evolution of the swarm density.



First, this thesis considers the case in which the agents' states evolve on a finite state space according to a continuous-time Markov chain (CTMC), and the forward equation is an ordinary differential equation (ODE). Defining the agents' task transition rates as the control parameters, the finite-time controllability, asymptotic controllability, and stabilization of the forward equation are investigated. Second, the controllability and stabilization problem for systems of advection-diffusion-reaction partial differential equations (PDEs) is studied in the case where the control parameters include the agents' velocity as well as transition rates. Third, this thesis considers a controllability and optimal control problem for the forward equation in the more general case where the agent dynamics are given by a nonlinear discrete-time control system. Beyond these theoretical results, this thesis also considers numerical optimal transport for control-affine systems. It is shown that finite-volume approximations of the associated PDEs lead to well-posed transport problems on graphs as long as the control system is controllable everywhere.

This dissertation presents the development of structural health monitoring and prognostic health management methodologies for complex structures and systems in the field of mechanical engineering. To overcome various challenges historically associated with complex structures and systems such as complicated sensing mechanisms, noisy information, and large-size datasets, a hybrid monitoring framework comprising of solid mechanics concepts and data mining technologies is developed. In such a framework, the solid mechanics simulations provide additional intuitions to data mining techniques reducing the dependence of accuracy on the training set, while the data mining approaches fuse and interpret information from the targeted system enabling the capability for real-time monitoring with efficient computation.

In the case of structural health monitoring, ultrasonic guided waves are utilized for damage identification and localization in complex composite structures. Signal processing and data mining techniques are integrated into the damage localization framework, and the converted wave modes, which are induced by the thickness variation due to the presence of delamination, are used as damage indicators. This framework has been validated through experiments and has shown sufficient accuracy in locating delamination in X-COR sandwich composites without the need of baseline information. Besides the localization of internal damage, the Gaussian process machine learning technique is integrated with finite element method as an online-offline prediction model to predict crack propagation with overloads under biaxial loading conditions; such a probabilistic prognosis model, with limited number of training examples, has shown increased accuracy over state-of-the-art techniques in predicting crack retardation behaviors induced by overloads. In the case of system level management, a monitoring framework built using a multivariate Gaussian model as basis is developed to evaluate the anomalous condition of commercial aircrafts. This method has been validated using commercial airline data and has shown high sensitivity to variations in aircraft dynamics and pilot operations. Moreover, this framework was also tested on simulated aircraft faults and its feasibility for real-time monitoring was demonstrated with sufficient computation efficiency.

This research is expected to serve as a practical addition to the existing literature while possessing the potential to be adopted in realistic engineering applications.

Aging-related damage and failure in structures, such as fatigue cracking, corrosion, and delamination, are critical for structural integrity. Most engineering structures have embedded defects such as voids, cracks, inclusions from manufacturing. The properties and locations of embedded defects are generally unknown and hard to detect in complex engineering structures. Therefore, early detection of damage is beneficial for prognosis and risk management of aging infrastructure system.

Non-destructive testing (NDT) and structural health monitoring (SHM) are widely used for this purpose. Different types of NDT techniques have been proposed for the damage detection, such as optical image, ultrasound wave, thermography, eddy current, and microwave. The focus in this study is on the wave-based detection method, which is grouped into two major categories: feature-based damage detection and model-assisted damage detection. Both damage detection approaches have their own pros and cons. Feature-based damage detection is usually very fast and doesn’t involve in the solution of the physical model. The key idea is the dimension reduction of signals to achieve efficient damage detection. The disadvantage is that the loss of information due to the feature extraction can induce significant uncertainties and reduces the resolution. The resolution of the feature-based approach highly depends on the sensing path density. Model-assisted damage detection is on the opposite side. Model-assisted damage detection has the ability for high resolution imaging with limited number of sensing paths since the entire signal histories are used for damage identification. Model-based methods are time-consuming due to the requirement for the inverse wave propagation solution, which is especially true for the large 3D structures.

The motivation of the proposed method is to develop efficient and accurate model-based damage imaging technique with limited data. The special focus is on the efficiency of the damage imaging algorithm as it is the major bottleneck of the model-assisted approach. The computational efficiency is achieved by two complimentary components. First, a fast forward wave propagation solver is developed, which is verified with the classical Finite Element(FEM) solution and the speed is 10-20 times faster. Next, efficient inverse wave propagation algorithms is proposed. Classical gradient-based optimization algorithms usually require finite difference method for gradient calculation, which is prohibitively expensive for large degree of freedoms. An adjoint method-based optimization algorithms is proposed, which avoids the repetitive finite difference calculations for every imaging variables. Thus, superior computational efficiency can be achieved by combining these two methods together for the damage imaging. A coupled Piezoelectric (PZT) damage imaging model is proposed to include the interaction between PZT and host structure. Following the formulation of the framework, experimental validation is performed on isotropic and anisotropic material with defects such as cracks, delamination, and voids. The results show that the proposed method can detect and reconstruct multiple damage simultaneously and efficiently, which is promising to be applied to complex large-scale engineering structures.

Robotic swarms can potentially perform complicated tasks such as exploration and mapping at large space and time scales in a parallel and robust fashion. This thesis presents strategies for mapping environmental features of interest – specifically obstacles, collision-free paths, generating a metric map and estimating scalar density fields– in an unknown domain using data obtained by a swarm of resource-constrained robots. First, an approach was developed for mapping a single obstacle using a swarm of point-mass robots with both directed and random motion. The swarm population dynamics are modeled by a set of advection-diffusion-reaction partial differential equations (PDEs) in which a spatially-dependent indicator function marks the presence or absence of the obstacle in the domain. The indicator function is estimated by solving an optimization problem with PDEs as constraints. Second, a methodology for constructing a topological map of an unknown environment was proposed, which indicates collision-free paths for navigation, from data collected by a swarm of finite-sized robots. As an initial step, the number of topological features in the domain was quantified by applying tools from algebraic topology, to a probability function over the explored region that indicates the presence of obstacles. A topological map of the domain is then generated using a graph-based wave propagation algorithm. This approach is further extended, enabling the technique to construct a metric map of an unknown domain with obstacles using uncertain position data collected by a swarm of resource-constrained robots, filtered using intensity measurements of an external signal. Next, a distributed method was developed to construct the occupancy grid map of an unknown environment using a swarm of inexpensive robots or mobile sensors with limited communication. In addition to this, an exploration strategy which combines information theoretic ideas with Levy walks was also proposed. Finally, the problem of reconstructing a two-dimensional scalar field using observations from a subset of a sensor network in which each node communicates its local measurements to its neighboring nodes was addressed. This problem reduces to estimating the initial condition of a large interconnected system with first-order linear dynamics, which can be solved as an optimization problem.

This paper describes an effort to bring wing structural stiffness and aeroelastic considerations early in the conceptual design process with an automated tool. Stiffness and aeroelasticity can be well represented with a stochastic model during conceptual design because of the high level of uncertainty and variability in wing non-structural mass such as fuel loading and control surfaces. To accomplish this, an improvement is made to existing design tools utilizing rule based automated design to generate wing torque box geometry from a specific wing outer mold-line. Simple analysis on deflection and inferred stiffness shows how early conceptual design choices can strongly impact the stiffness of the structure. The impacts of design choices and how the buckling constraints drive structural weight in particular examples are discussed. The model is then carried further to include a finite element model (FEM) to analyze resulting mode shapes and frequencies for use in aeroelastic analysis. The natural frequencies of several selected wing torque boxes across a range of loading cases are compared.

Mistuning is defined as the blade-to-blade variation of bladed disks caused by slight changes in material or geometric properties; mistuned blades can cause significant increases in vibrational response. The primary goal of this thesis is to describe the relationship between coupling index and amplification factors of mistuned bladed disks with various sets of parameters, targeting the veering zone. At around a coupling index of 0, the amplification factors tend to stay around 1. This is due to localization of energy, where no energy is "shared" between blades, and the response of mistuned blades remain at resonance. As coupling index increases, amplification factors reach a peak between coupling indices of 0.15 and 0.2, before experiencing a downward trend towards 1. As blade-to-disk interaction increases, more energy is "shared" across blades. This results in the upward trend of amplification factor as coupling index increases, until too much energy is "shared". Additionally, a reduced order model enriching-stripping process to match natural frequencies of Nastran simulations will be discussed. This thesis is a continuation of Saurav Sahoo's Master's thesis at Arizona State University, Approximate a-priori Estimation of the Response Amplification due to Geometric and Young's Modulus Mistuning.