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About this book

This multidisciplinary volume features invited contributions on mathematical applications in naval engineering. Seeking a more holistic approach that transcends current scientific boundaries, leading experts present interdisciplinary instruments and models on a broad range of topics. Each chapter places special emphasis on important methods, research directions, and applications of analysis within the field. Fundamental scientific and mathematical concepts are applied to topics such as the butterfly structure of the FFT, the acoustic impedance of pistons in a two-layer medium, deterministic batch trackers, spline equations, moving horizons estimation, membership games for planning sensor networks, statistical models of inertial sensors, random flight searches in bounded domains, the acoustics of a mixed porosity felt airfoil, and a novel aft boundary condition for a towed flexible cylinder.

Carefully peer-reviewed and pedagogically presented for a broad readership, this volume is perfect for graduate and postdoctoral students interested in interdisciplinary research. Researchers in applied mathematics and sciences will find this book an important resource on the latest developments in the field. In keeping with the STEAM-H series, this volume hopes to inspire interdisciplinary understanding and collaboration.

Table of Contents


Impedance of Pistons on a Two–Layer Medium with Inviscid Homogeneous Flow

An integral transform technique is used to develop a general solution for the impedance of two-dimensional pistons acting on a two-layer medium. The medium consists of a semi-infinite fluid with homogeneous subsonic flow above a viscoelastic layer in a rigid infinite baffle. The stresses acting on the planar baffle and piston, as a result of piston motion, are determined using linear elasticity theory and the pressures in the fluid are determined using the convected wave equation. The case of a rigid piston of length L is considered. The derived expression for impedance is evaluated by direct numerical integration of the wavenumber transformed solution. Numerical results over a range of flow speeds and layer thicknesses are compared with classical impedance functions. At low frequencies (k0L < 2), the impedances vary significantly from the classical piston impedance functions due to the shear properties of the viscoelastic medium and flow Mach number. These deviations from the classical piston impedance occur for both resistance and reactance. It was also found that with increasing Mach number, the reactive component of impedance transitions from negative (spring-like) to positive (mass-like) in the low frequency range. At higher frequencies, the influence of Mach number is secondary to layer thickness.
Scott E. Hassan

Acoustics of a Mixed Porosity Felt Airfoil

Lifting surfaces produce noise in operation, and porous materials have been shown to reduce this noise. Acoustic measurements were made on three arrangements of poroelastic material: full-chord impermeability, full-chord porosity, and “mixed” porosity. The results are compared to applicable published data. The noise reduction produced by the mixed porosity arrangement is found to be similar to the fully porous arrangement at low Reynolds numbers. The noise reduction produced by the mixed and fully porous foils shows significant differences at higher Reynolds numbers. A percolation-base physical model to explain elevated noise production at high frequencies is proposed.
Aren M. Hellum

Generalizing the Butterfly Structure of the FFT

The structure of the various forms of the fast Fourier transform (FFT) is well described by patterns of “butterfly” operations, each involving only an individual pair of inputs or intermediate results, but ultimately yielding one of the most elegant, efficient, and ubiquitous computational algorithms known to mathematics. Here, the structure underlying the FFT is developed in a more general context of a decomposition applicable to any arbitrary complex unitary matrix. By developing this multi-layer decomposition (MLD), it becomes clear that:
  • Any unitary matrix may be decomposed into a butterfly flow graph.
  • Each individual butterfly may be interpreted as an equivalent 2 × 2 standard complex Givens rotation.
  • Identification of the butterfly flow graph for any particular unitary matrix is intimately related to the singular value decomposition (SVD) of the different block sub-matrices of the original unitary matrix.
  • The resulting butterfly flow graph may be interpreted as a prescription of the individual planar coordinate rotations necessary to precisely implement the general multi-dimensional coordinate rotation defined by the unitary matrix. It may also be interpreted as the equivalent polar representation for the matrix.
  • The general form of the butterfly flow graph is not “fast.” The FFT achieves its computational efficiency due to certain specific mathematical symmetries, permitting removal of extraneous butterflies.
  • The form of the butterfly flow graph is sensitive to the order in which the inputs and outputs are specified.
  • A distinct advantage of the MLD is that any approximations or modifications made within this structure continue to result in a full unitary transformation.
An initial MATLAB code set implementing MLD is provided. The author believes the basic MLD concept to be a novel procedure which opens the possibility for subsequent exploitation in a number of different directions.
John Polcari

Development of an Aft Boundary Condition for a Horizontally Towed Flexible Cylinder

The method of characteristics is used to develop an aft boundary condition for the linear transverse dynamics of a towed neutrally buoyant flexible cylinder, extending a previous approach to more general numerical methods (e.g., finite differences, finite elements, etc.). The boundary condition is located at the critical point on the cylinder (i.e., where the tension and hydrodynamic forces balance) instead of at the free end. This avoids the region where bending terms become important, and it removes the problem of modeling the instabilities in the region near the free end. The approach can also be extended to the nonlinear case.
Anthony A. Ruffa

Tracking with Deterministic Batch Trackers

In this chapter, we present two deterministic (i.e., non-Bayesian) batch trackers—the Maximum Likelihood Probabilistic Data Association (ML-PDA) tracker and the Maximum Likelihood Probabilistic Multi-Hypothesis Tracker (ML-PMHT). Both trackers formulated by making assumptions about the target and the environment in which the target is present. Using these assumptions, in both cases, a log-likelihood ratio (LLR) is formulated, and then the state vector x that maximizes this LLR is usually chosen as the target state. Both the ML-PDA and the ML-PMHT LLRs are developed. We specifically consider two different amplitude likelihood ratios that have been used in these trackers—a fluctuating Gaussian model and a heavier-tailed clutter model. Finally, we present a method for determining a “tracking threshold” for ML-PMHT—i.e., if the maximum ML-PMHT likelihood value for a batch of data is above this level, it is determined to be a target; if it is below this level, the peak is rejected as clutter originated.
Steven Schoenecker

Moving Horizons Estimation for Wheelchair Trajectory Repeatability in the Home

An advantage of Moving Horizons Estimation in contrast with the previously tested Extended Kalman Filter is presented in the context of achieving a useful form of teach/repeat for wheelchairs of severely disabled veterans within their homes. The ability to combine numerical integrals of the measured wheel rotations of the chair with chair-mounted cameras and camera-based observations of wall-mounted fiducials within a trailing, selected “window” based on the “taught” trajectory, in order to precisely and reliably repeat that trajectory, is presented. The importance and real-time feasibility of the observation-batch-selection protocol—which is applied both to the teaching data and, in real time, to the tracking data—is discussed. Experimental illustration uses the trajectory shown in https://​youtu.​be/​7Yrc3IuXBus.
Steven B. Skaar

Exact Solutions to the Spline Equations

The exact solution to the cubic spline equations is developed for the case of equal knot spacing. It exhibits an oscillatory response in the region of a discontinuity, which is a consequence of the row structure of the resulting tridiagonal Toeplitz system. The oscillations cancel in the absence of discontinuities. Splines under tension exhibit a similar oscillatory response; however, increasing the tension attenuates the oscillations over a shorter length scale. The use of imaginary tension removes the large amplitude oscillations in the region of a discontinuity at the expense of introducing a low-amplitude oscillation throughout the entire curve fit. A composite spline (i.e., a spline under tension in the region surrounding a discontinuity, and a cubic spline elsewhere) can confine such oscillations to an arbitrarily defined region surrounding each discontinuity.
Anthony A. Ruffa, Bourama Toni

Distributed Membership Games for Planning Sensor Networks

We consider the management of sensor networks which are organized into distinct groups that share data to improve performance. In particular, for systems where changes in individual sensors’ group membership may improve performance (in response to local conditions and/or group performance), the requirement of control by a central authority may make such adaptation prohibitive. We develop two different group membership game formulations to address this issue. The first type of game we develop is a group affiliation game that creates joining rules based on perceived maximal group size. The second type of game is a group allocation game of distributed welfare where the sensors work noncooperatively to self-organize into a beneficial distribution of group memberships. We derive group joining rules for individual sensors in each game formulation and illustrate the results with numerical calculations.
Thomas A. Wettergren, C. Michael Traweek

Statistical Models of Inertial Sensors and Integral Error Bounds

Inertial sensors such as gyroscopes and accelerometers are important components of inertial measurement units (IMUs). Sensor output signals are corrupted by additive noise plus a random drift component. This drift component, also called bias, is modeled using different types of random processes. This chapter considers the random components that are useful for modeling modern tactical-graded MEMS sensors. These components contribute to errors in the first and second integrals of the sensor output. The main contribution of this chapter is the derivation of a statistical bound on the magnitude of the error in the integral of a sensor signal due to noise and drift. This bound is a simple function of the Allan variance of a sensor.
Richard J. Vaccaro, Ahmed S. Zaki

Developing Efficient Random Flight Searches in Bounded Domains

We develop a new method for computing the parameters defining an efficient random flight for searchers that are constrained to search in a bounded domain. Using concepts from observations of animal foraging behavior, we define a random search plan that provides an optimally efficient search in terms of coverage relative to the constraints of random motion in the bounded domain. In contrast to the previous studies, our method directly accounts for the change to the behavior that occurs for bounded flights. In addition, we show how to modify the resulting random flight parameters to account for the effects of multiple searchers in the same region. Numerical simulations are performed to illustrate the effectiveness of this random search strategy.
Thomas A. Wettergren


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