Advances in Interdisciplinary Mathematical Research

The NASA Langley Research Center’s Research Directorate provides many of the research and technology development capabilities required by the present and future needs of NASA across three encompassing technology areas, namely, aerodynamics, aerothermodynamics and acoustics (AAA); structures and materials (SM); and Airborne Systems (AirSc). Researchers contribute to nine primary areas of expertise which include structures, hypersonics, materials, flight dynamics and control, measurement sciences, crew systems and aviation operations, aerodynamics, safety critical avionics systems, and acoustics. These areas of expertise cover virtually all of the important disciplines related to flight, including the agency’s main thrusts within structures and materials. Researchers in the structures and materials technology area are constantly working to develop advanced materials to enable efficient, high-performance aerospace concepts; efficient, physics-based analytical and computational methods for multidisciplinary design and analysis; and methods to quantify the behavior, durability, damage tolerance, and overall performance of advanced materials and structures.As part of the structures and materials technology area, the Durability, Damage Tolerance and Reliability Branch (DDTRB) conducts research and technology development of efficient, physics-based analytical and computational methods to enable multidisciplinary design and analysis of advanced materials and structures for aerospace applications, including evaluation of concepts, quantification of behavior, durability, and damage tolerance, and validation of performance.DDTRB has contributed to the development and implementation of many fracture mechanics methods aimed at predicting and characterizing damage in both metallic and composite materials. Engineering fracture mechanics plays a vital role in the development and certification of virtually every aerospace vehicle that has been developed since the mid-twentieth century. This chapter presents a selection of computational, analytical, and experimental strategies and methodologies that have been developed by the branch for simulating and assessing damage growth under monotonic and cyclic loading and for characterizing the damage tolerance of aerospace structures. It includes continuum-based mechanics as well as a new paradigm focused on simulating and characterizing fundamental damage processes, called damage science.

The \(\Gamma \) -convergence theory deals with a singular limit phenomenon in the calculus of variations. It provides a rigorous notion for a family of functionals to converge to a functional of seemingly a different type, while still retaining vital properties in the limiting functional. For instance, global minimizers of the functionals in the family converge to a global minimizer of the limiting functional. Near an isolated local minimizer of the limiting functional, there exist local minimizers of the functionals in the converging family that are sufficiently close to the limiting functional. This theory has found a surprising application in the study of block copolymer morphology. Block copolymers are soft materials characterized by fluid-like disorder on the molecular scale and a high degree of order at a longer length scale. This chapter presents a description of the Ohta–Kawasaki theory density theory for block copolymer morphology and applies the \(\Gamma \) -convergence theory to reduce the Ohta–Kawasaki theory to a geometric problem containing perimeter minimization and nonlocal interaction. As an application of \(\Gamma \) -convergence, one determines all the global and local minimizers in one dimension. Consequently global and local minimizers are also characterized for the Ohta–Kawasaki theory.

This chapter represents an attempt to summarize some of the direct and indirect connections that exist between ray theory, wave theory, and potential scattering theory. Such connections have been noted in the past and have been exploited to some degree, but in the opinion of this author, there is much more yet to be pursued in this regard. This article provides the framework for more detailed analysis in the future. In order to gain a better appreciation for a topic, it is frequently of value to examine it from as many complementary levels of description as possible, and that is the objective here. Drawing in part on the work of Nussenzveig, Lock, Debye, and others, the mathematical nature of the rainbow is discussed from several perspectives. The primary bow is the lowest-order bow that can occur by scattering from a spherical drop with constant refractive index n, but zero-order (or direct transmission) bows can exist when the sphere is radially inhomogeneous. The refractive index profile automatically defines a scattering potential but with a significant difference compared to the standard quantum mechanical form: the potential is k-dependent. A consequence of this is that there are no bound states for this system. The correspondences between the resonant modes in scattering by a potential of the “well-barrier” type and the behavior of electromagnetic “rays” in a transparent (or dielectric) sphere are discussed. The poles and saddle points of the associated scattering matrix have quite profound connections to electromagnetic tunneling, resonances, and “rainbows” arising within and from the sphere. The links between the various mathematical and physical viewpoints are most easily appreciated in the case of constant n, thus providing insight into possible extensions to these descriptions for bows of arbitrary order in radially inhomogeneous spheres (and cylinders).

Although rare, collisions of two or more bodies in the N-body problem are apparent obstacles at which Newton’s law of gravity ceases to make sense. Without understanding the nature of collisions, a complete understanding of the N-body problem cannot be achieved. Historically, several methods have been developed to probe the nature of collisions in the N-body problem. One of these methods removes, or regularizes, certain types of collisions entirely, thereby relating not only analytically but also numerically the dynamics of such collision motions with their near-collision motions. By understanding the dynamics of collision motions in the regularized setting, a better understanding of the dynamics of near-collision motions is achieved.

Some recent results for analyzing the stability of equilibrium of delay differential equations are reviewed. Systems of one or two equations in general form are considered, and the criterions for absolute stability or conditional stability are given explicitly. The results show how the stability depends on both the instantaneous feedback and the delayed feedback.

We are concerned in this paper with the antiperiodicity of mild solutions for the semilinear evolution equation \(x^{\prime}(t) = Ax(t) + f(t,x)\)where A is a sectorial operator not necessarily densely defined in X generating an hyperbolic semigroup \((T(t))_{t\geq 0}\)in a Banach space X and \(f : \mathbb{R} \times X_{\alpha } \rightarrow X\), where X _α is an intermediate space. We prove the existence and uniqueness of an antiperiodic mild solution in X _α , when the function \(f : \mathbb{R} \times X_{\alpha }\rightarrow X\)is antiperiodic. The result is obtained using the Banach-fixed point theorem.

Computer-aided decision-making systems have been introduced into many fields, such as economics, medicine, architecture, and agriculture. The increasing demand and rapid pace of development of such computer-aided decision-making systems displays their popularity and success in aiding and enhancing various fields. In the field of medicine, the advantage of having such systems is in the expense, labor, energy, and budget savings they provide to the health care environments. In the following sections, a brief description of the application of such systems in hemorrhagic shock, attention detection, traumatic brain injuries, and pelvic fracture detection has been provided. A flowchart of the procedure of developing such systems is represented in Fig. 7.1.

In general, most acquired datasets suffer from the effects of acquisition noise, channel noise, fading, and fusion nodes. At the decision-making stage, where these data are fused together, any deviation from the real values of these data could affect the decisions made. We use a wavelet transform approach to develop computationally low-power, low bandwidth, and low-cost filters that will remove the noise effectively so that a decision can be made at the node level. This wavelet-based method is guaranteed to converge to a stationary point for both uncorrelated and correlated datasets. Presented here is an overview of the theoretical background illustrated with some experimental results showing the performance and merits of this novel approach.

In this article, we study the null-controllability problem with two constraints on a pair of controls.We apply these results to a discriminating sentinel with given sensitivity to detect some parameters in a pollution problem, governed by a semi-linear parabolic equation with Dirichlet boundary condition.

A heat transfer model of fluid flow inside closed channels with arbitrary shapes has been built from momentum and energy equations. The model equations have been solved with Galerkin-based method. Detailed velocity and temperature fields in the fluid flow have been obtained. Using the solution fields the friction factors and heat transfer rates have been also calculated. To validate the developed solution procedure, the results have been compared to the results of numerical methods and experimental data.

We extend the theories of limit cycles and quasi-periodicity to the new concepts of almost and pseudo-almost limit cycles. We investigate the conditions of existence, uniqueness, and stability and introduce the notion of almost and pseudo isochrons of almost and pseudo-limit cycles. To illustrate we present several examples including some almost and pseudo-almost periodic perturbations of the harmonic oscillator and the renowned Liénard systems along with their graphic requirements. We derive the existence of almost and pseudo-almost periodic waves by perturbing first then transforming some hyperbolic and parabolic partial differential equations to Liénard-type equations. Included are some open questions on the co-existence of limit cycles and strictly almost/pseudo-almost limit cycles, the accumulation of almost/pseudo-almost limit cycles, and the bifurcation of almost/pseudo-almost limit cycles in parameterized systems, the existence of isochronous almost and pseudo-almost limit cycles. Finally through coupling and synchronization of almost or pseudo-almost self-sustained oscillators, and possibly their almost or pseudo-almost isochrons, we conjecture the transition from strictly almost/pseudo-almost periodic behavior to chaotic behavior.

In this paper almost periodic random sequence in mean is defined and investigated. It is then applied to study the existence and uniqueness of the almost periodic solution of a semi-linear system of stochastic difference equations of the form: by means of exponential dichotomy.