A Characteristic Dynamic Mode Decomposition

Temporal or spatial structures are readily extracted from complex data by modal decompositions like Proper Orthogonal Decomposition (POD) or Dynamic Mode Decomposition (DMD). Subspaces of such decompositions serve as reduced order models and define either spatial structures in time or temporal structures in space. On the contrary, convecting phenomena pose a major problem to those decompositions. A structure traveling with a certain group velocity will be perceived as a plethora of modes in time or space respectively. This manifests itself for example in poorly decaying singular values when using a POD. The poor decay is counter-intuitive, since a single structure is expected to be represented by a few modes. The intuition proves to be correct and we show that in a properly chosen reference frame along the characteristics defined by the group velocity, a POD or DMD reduces moving structures to a few modes, as expected. Beyond serving as a reduced model, the resulting entity can be used to define a constant or minimally changing structure in turbulent flows. This can be interpreted as an empirical counterpart to exact coherent structures. We present the method and its application to a head vortex of a compressible starting jet

The Geometry of Regular Shear-Free Null Geodesic Congruences, CR functions and their Application to the Flat-Space Maxwell Equations

We describe here what appears to be a new structure that is hidden in all asymptotically vanishing Maxwell fields possessing a non-vanishing total charge. Though we are dealing with real Maxwell fields on real Minkowski space nevertheless, directly from the asymptotic field one can extract a complex analytic world-line defined in complex Minkowski space that gives a unified Lorentz invariant meaning to both the electric and magnetic dipole moments. In some sense the world-line defines a `complex center of charge' around which both electric and magnetic dipole moments vanish. The question of how and where does this complex world-line arise is one of the two main subjects of this work. The other subject concerns what is known in the mathematical literature as a CR structure. In GR, CR structures naturally appear in the physical context of shear-free (or asymptotically shear-free) null geodesic congruences in space-time. For us, the CR structure is associated with the embedding of Penrose's real three-dimensional null infinity, I^+, as a surface in a two complex dimensional space, C^2. It is this embedding, via a complex function, (a CR function), that is our other area of interest. Specifically we are interested in the `decomposition' of the CR function into its real and imaginary parts and the physical information contained in this decomposition.Comment: 25 page

Guided Wave Based Integrated Structural Health Monitoring and Nondestructive Evaluation

Damage detection and health monitoring are critical for ensuring the structural safety in various fields, such as aerospace, civil and nuclear engineering. Structural health monitoring (SHM) performs online nondestructive evaluation (NDE) and can predict the structural remaining life through appropriate diagnosis and prognosis technologies. Among various SHM/NDE technologies, guided ultrasonic waves have shown great potential for fast and large area SHM/NDE, due to their sensitivity to small defects and capability to propagate long distances. Recent advances in guided wave based SHM/NDE technologies have demonstrated the feasibility of detecting damage in simple structures such as metallic plates and pipes. However, there remain many challenging tasks for quantifying damage, especially for damage quantification in complex structures such as laminated composites and honeycomb sandwich structures. Moreover, guided wave propagations in complex structures, and wave interactions with various types of defects such as crack, delamination and debonding damage, need to be investigated. The objective of this dissertation research is to develop guided wave based integrated SHM and NDE methodologies for damage detection and quantification in complex structures. This objective is achieved through guided wave modeling, optimized sensor and sensing system development, and quantitative and visualized damage diagnoses. Moreover, the developed SHM/NDE methodologies are used for various damage detection and health monitoring applications. This dissertation is organized in two major parts. Part I focuses on the development of integrated SHM/NDE damage diagnosis methodologies. A non-contact laser vibrometry sensing system is optimized to acquire high spatial resolution wavefields of guided waves. The guided wavefields in terms of time and space dimensions contain a wealth of information regarding guided wave propagations in structures and wave interactions with structural discontinuities. To extract informative wave signatures from the time-space wavefields and characterize the complex wave propagation and interaction phenomenon, guided wavefield analysis methods, including frequency-wavenumber analysis, wavefield decomposition and space-frequency-wavenumber analysis, are investigated. Using these analysis methods, the multi-modal and dispersive guided waves can be resolved, and the complex wave propagation and interaction can be interpreted and analyzed in time, space, frequency, and wavenumber multi-domains. In Part I, a hierarchical damage diagnosis methodology is also developed for quantitative and visualized damage detection. The hierarchical methodology systematically combines phased array imaging and wavefield based imaging to achieve efficient and precise damage detection and quantification. The generic phased array imaging is developed based on classic delay-and-sum principle and works for both isotropic and anisotropic materials. Using the phased array imaging, an intensity scanning image of the structure is generated to efficiently visualize and locate the damage zone. Then the wavefield based imaging methods such as filter reconstruction imaging and spatial wavenumber imaging are performed to precisely quantify the damage size, shape and depth. In Part II, the developed methodologies are applied to five different SHM/NDE applications: (1) gas accumulation detection and quantification in water loaded structures, (2) crack damage detection and quantification in isotropic plates, (3) thickness loss evaluation in isotropic plates, (4) delamination damage detection and quantification in composite laminates, (5) debonding detection and quantification in honeycomb sandwich structures. This dissertation research will initiate sensing and diagnosis methodologies that provide rapid noncontact inspection of damage and diagnosis of structural health. In the long run, it contributes to the development of advanced sensor and sensing technologies based on guided waves, and to providing on-demand health information at component or subsystem level for the safety and reliability of the structure

Cusped hyperbolic 3-manifolds: canonically CAT(0) with CAT(0) spines

We prove that every finite-volume hyperbolic 3-manifold M with p > 0 cusps admits a canonical, complete, piecewise Euclidean CAT(0) metric, with a canonical projection to a CAT(0) spine K. Moreover, (a) the universal cover of M endowed with the CAT(0) metric is a union of Euclidean half-spaces, glued together by identifying Euclidean polygons in their bounding planes by pairwise isometry (b)each cusp of M in the CAT(0) metric is a non-singular metric product of a (Euclidean) cusp torus and a half-line (c) all metric singularities are concentrated on the 1-skeleton of K, with cone angles a multiple of pi (d) there is a canonical deformation of the hyperbolic metric with limit the CAT(0) piecewise Euclidean metric. The proof uses Ford domains; the construction is essentially the polar-dual of the Epstein-Penner canonical decomposition, and generalizes to higher dimension.Comment: 16 pages, 3 figure

Spin structures on compact homogeneous pseudo-Riemannian manifolds

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic structure. Then we investigate spin structures on principal torus bundles over flag manifolds, i.e. C-spaces, or equivalently simply-connected homogeneous complex manifolds M=G/L of a compact semisimple Lie group G. We study the topology of M and we provide a sufficient and necessary condition for the existence of an (invariant) spin structure, in terms of the Koszul form of F. We also classify all C-spaces which are fibered over an exceptional spin flag manifold and hence they are spin

Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables

Understanding nonlinear dynamical systems (NLDSs) is challenging in a variety of engineering and scientific fields. Dynamic mode decomposition (DMD), which is a numerical algorithm for the spectral analysis of Koopman operators, has been attracting attention as a way of obtaining global modal descriptions of NLDSs without requiring explicit prior knowledge. However, since existing DMD algorithms are in principle formulated based on the concatenation of scalar observables, it is not directly applicable to data with dependent structures among observables, which take, for example, the form of a sequence of graphs. In this paper, we formulate Koopman spectral analysis for NLDSs with structures among observables and propose an estimation algorithm for this problem. This method can extract and visualize the underlying low-dimensional global dynamics of NLDSs with structures among observables from data, which can be useful in understanding the underlying dynamics of such NLDSs. To this end, we first formulate the problem of estimating spectra of the Koopman operator defined in vector-valued reproducing kernel Hilbert spaces, and then develop an estimation procedure for this problem by reformulating tensor-based DMD. As a special case of our method, we propose the method named as Graph DMD, which is a numerical algorithm for Koopman spectral analysis of graph dynamical systems, using a sequence of adjacency matrices. We investigate the empirical performance of our method by using synthetic and real-world data.Comment: 34 pages with 4 figures, Published in Neural Networks, 201

The Stellar tree: a Compact Representation for Simplicial Complexes and Beyond

We introduce the Stellar decomposition, a model for efficient topological data structures over a broad range of simplicial and cell complexes. A Stellar decomposition of a complex is a collection of regions indexing the complex's vertices and cells such that each region has sufficient information to locally reconstruct the star of its vertices, i.e., the cells incident in the region's vertices. Stellar decompositions are general in that they can compactly represent and efficiently traverse arbitrary complexes with a manifold or non-manifold domain They are scalable to complexes in high dimension and of large size, and they enable users to easily construct tailored application-dependent data structures using a fraction of the memory required by the corresponding topological data structure on the global complex. As a concrete realization of this model for spatially embedded complexes, we introduce the Stellar tree, which combines a nested spatial tree with a simple tuning parameter to control the number of vertices in a region. Stellar trees exploit the complex's spatial locality by reordering vertex and cell indices according to the spatial decomposition and by compressing sequential ranges of indices. Stellar trees are competitive with state-of-the-art topological data structures for manifold simplicial complexes and offer significant improvements for cell complexes and non-manifold simplicial complexes. As a proxy for larger applications, we describe how Stellar trees can be used to generate existing state-of-the-art topological data structures. In addition to faster generation times, the reduced memory requirements of a Stellar tree enable generating these data structures over large and high-dimensional complexes even on machines with limited resources

The behaviour of Fenchel-Nielsen distance under a change of pants decomposition

Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition $\mathcal{P}$ and given a base complex structure $X$ on $S$, there is an associated deformation space of complex structures on $S$, which we call the Fenchel-Nielsen Teichm\"uller space associated to the pair $(\mathcal{P},X)$. This space carries a metric, which we call the Fenchel-Nielsen metric, defined using Fenchel-Nielsen coordinates. We studied this metric in the papers \cite{ALPSS}, \cite{various} and \cite{local}, and we compared it to the classical Teichm\"uller metric (defined using quasi-conformal mappings) and to another metric, namely, the length spectrum, defined using ratios of hyperbolic lengths of simple closed curves metric. In the present paper, we show that under a change of pair of pants decomposition, the identity map between the corresponding Fenchel-Nielsen metrics is not necessarily bi-Lipschitz. The results complement results obtained in the previous papers and they show that these previous results are optimal

Dynamics of random pressure fields over bluff bodies: a dynamic mode decomposition perspective

Aerodynamic pressure field over bluff bodies immersed in boundary layer flows is correlated both in space and time. Conventional approaches for the analysis of distributed aerodynamic pressures, e.g., the proper orthogonal decomposition (POD), can only offer relevant spatial patterns in a set of coherent structures. This study provides an operator-theoretic approach that describes dynamic pressure fields in a functional space rather than conventional phase space via the Koopman operator. Subsequently, spectral analysis of the Koopman operator provides a spatiotemporal characterization of the pressure field. An augmented dynamic mode decomposition (DMD) method is proposed to perform the spectral decomposition. The augmentation is achieved by the use of the Takens's embedding theorem, where time delay coordinates are considered. Consequently, the identified eigen-tuples (eigenvalues, eigenvectors, and time evolution) can capture not only dominant spatial structures but also identify each structure with a specific frequency and a corresponding temporal growth/decay. This study encompasses learning the evolution dynamics of the random aerodynamic pressure field over a scaled model of a finite height prism using limited wind tunnel data. The POD analysis of the experimental data was also carried out. To demonstrate the unique feature of the proposed approach, the DMD and POD based learning results including algorithm convergence, data sufficiency, and modal analysis are examined. The ensuing observations offer a glimpse of the complex dynamics of the surface pressure field over bluff bodies that lends insights to features previously masked by conventional analysis approaches.Comment: 32 pages, 24 figure

Entanglement entropy of squeezed vacua on a lattice

We derive a formula for the entanglement entropy of squeezed states on a lattice in terms of the complex structure J. The analysis involves the identification of squeezed states with group-theoretical coherent states of the symplectic group and the relation between the coset Sp(2N,R)/Isot(J_0) and the space of complex structures. We present two applications of the new formula: (i) we derive the area law for the ground state of a scalar field on a generic lattice in the limit of small speed of sound, (ii) we compute the rate of growth of the entanglement entropy in the presence of an instability and show that it is bounded from above by the Kolmogorov-Sinai rate.Comment: 35 pages, 2 figure