GRID2D/3D: A computer program for generating grid systems in complex-shaped two- and three-dimensional spatial domains. Part 1: Theory and method

An efficient computer program, called GRID2D/3D was developed to generate single and composite grid systems within geometrically complex two- and three-dimensional (2- and 3-D) spatial domains that can deform with time. GRID2D/3D generates single grid systems by using algebraic grid generation methods based on transfinite interpolation in which the distribution of grid points within the spatial domain is controlled by stretching functions. All single grid systems generated by GRID2D/3D can have grid lines that are continuous and differentiable everywhere up to the second-order. Also, grid lines can intersect boundaries of the spatial domain orthogonally. GRID2D/3D generates composite grid systems by patching together two or more single grid systems. The patching can be discontinuous or continuous. For continuous composite grid systems, the grid lines are continuous and differentiable everywhere up to the second-order except at interfaces where different single grid systems meet. At interfaces where different single grid systems meet, the grid lines are only differentiable up to the first-order. For 2-D spatial domains, the boundary curves are described by using either cubic or tension spline interpolation. For 3-D spatial domains, the boundary surfaces are described by using either linear Coon's interpolation, bi-hyperbolic spline interpolation, or a new technique referred to as 3-D bi-directional Hermite interpolation. Since grid systems generated by algebraic methods can have grid lines that overlap one another, GRID2D/3D contains a graphics package for evaluating the grid systems generated. With the graphics package, the user can generate grid systems in an interactive manner with the grid generation part of GRID2D/3D. GRID2D/3D is written in FORTRAN 77 and can be run on any IBM PC, XT, or AT compatible computer. In order to use GRID2D/3D on workstations or mainframe computers, some minor modifications must be made in the graphics part of the program; no modifications are needed in the grid generation part of the program. This technical memorandum describes the theory and method used in GRID2D/3D

A zonal computational procedure adapted to the optimization of two-dimensional thrust augmentor inlets

A viscous-inviscid interaction methodology based on a zonal description of the flowfield is developed as a mean of predicting the performance of two-dimensional thrust augmenting ejectors. An inviscid zone comprising the irrotational flow about the device is patched together with a viscous zone containing the turbulent mixing flow. The inviscid region is computed by a higher order panel method, while an integral method is used for the description of the viscous part. A non-linear, constrained optimization study is undertaken for the design of the inlet region. In this study, the viscous-inviscid analysis is complemented with a boundary layer calculation to account for flow separation from the walls of the inlet region. The thrust-based Reynolds number as well as the free stream velocity are shown to be important parameters in the design of a thrust augmentor inlet

The rolling problem: overview and challenges

In the present paper we give a historical account -ranging from classical to modern results- of the problem of rolling two Riemannian manifolds one on the other, with the restrictions that they cannot instantaneously slip or spin one with respect to the other. On the way we show how this problem has profited from the development of intrinsic Riemannian geometry, from geometric control theory and sub-Riemannian geometry. We also mention how other areas -such as robotics and interpolation theory- have employed the rolling model.Comment: 20 page

Positive approximations of the inverse of fractional powers of SPD M-matrices

This study is motivated by the recent development in the fractional calculus and its applications. During last few years, several different techniques are proposed to localize the nonlocal fractional diffusion operator. They are based on transformation of the original problem to a local elliptic or pseudoparabolic problem, or to an integral representation of the solution, thus increasing the dimension of the computational domain. More recently, an alternative approach aimed at reducing the computational complexity was developed. The linear algebraic system $\cal A^\alpha \bf u=\bf f$, $0< \alpha <1$ is considered, where $\cal A$ is a properly normalized (scalded) symmetric and positive definite matrix obtained from finite element or finite difference approximation of second order elliptic problems in $\Omega\subset\mathbb{R}^d$, $d=1,2,3$. The method is based on best uniform rational approximations (BURA) of the function $t^{\beta-\alpha}$ for $0 < t \le 1$ and natural $\beta$. The maximum principles are among the major qualitative properties of linear elliptic operators/PDEs. In many studies and applications, it is important that such properties are preserved by the selected numerical solution method. In this paper we present and analyze the properties of positive approximations of $\cal A^{-\alpha}$ obtained by the BURA technique. Sufficient conditions for positiveness are proven, complemented by sharp error estimates. The theoretical results are supported by representative numerical tests

Wavelet and Multiscale Methods

Various scientific models demand finer and finer resolutions of relevant features. Paradoxically, increasing computational power serves to even heighten this demand. Namely, the wealth of available data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information leads to tasks that are not tractable by standard numerical techniques. The last decade has seen the emergence of several new computational methodologies to address this situation. Their common features are the nonlinearity of the solution methods as well as the ability of separating solution characteristics living on different length scales. Perhaps the most prominent examples lie in multigrid methods and adaptive grid solvers for partial differential equations. These have substantially advanced the frontiers of computability for certain problem classes in numerical analysis. Other highly visible examples are: regression techniques in nonparametric statistical estimation, the design of universal estimators in the context of mathematical learning theory and machine learning; the investigation of greedy algorithms in complexity theory, compression techniques and encoding in signal and image processing; the solution of global operator equations through the compression of fully populated matrices arising from boundary integral equations with the aid of multipole expansions and hierarchical matrices; attacking problems in high spatial dimensions by sparse grid or hyperbolic wavelet concepts. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computation and to promote the exchange of ideas emerging in various disciplines

Synthesis of mechanisms for function, path, and motion generation using invariant characterization, storage and search methods

This work presents an approach for the synthesis of four-bar planar mechanisms for function, path, and motion generation based upon the use of invariant descriptors and local database generation and search methods;Transformation and descriptor methods are used to characterize the motion of four-bar planar mechanisms (function, path and motion generation) and to store the invariant characteristic information in a database. Spatial transforms, one-dimensional Fourier transforms, two-dimensional Fourier transforms, and invariant moments are used to generate invariant characteristic descriptors. The resulting characteristic information for each curve is invariant regardless of the rotation, translation or scaling of the curves. A description of each method and the relative performance of file development and search methods are developed. Over 8,000 function, path and motion solutions are generated for global search solutions for each transform and descriptor method. The function, path and motion solutions are based on the solutions developed by Hrones and Nelson and based on the implementation of a random search of a local design space. Solution comparison and matching techniques are discussed, and implemented, to evaluate the deviation of a candidate curve to curves stored in a database;A methodology is developed to allow the designer to investigate a local design space by generating a database of candidate solutions based on the random development of four-bar mechanisms. The designer may then define a desired solution and search the generated candidate solution files. This technique supports the evaluation of a local solution space through the generation, characterization, and identification of candidate mechanisms that may be practical to implement. After the identification of candidate mechanisms, local optimization techniques may be used with candidate mechanisms

Sampling and Reconstruction of Spatial Signals

Digital processing of signals f may start from sampling on a discrete set Γ, f →(f(ϒη))ϒηεΓ. The sampling theory is one of the most basic and fascinating topics in applied mathematics and in engineering sciences. The most well known form is the uniform sampling theorem for band-limited/wavelet signals, that gives a framework for converting analog signals into sequences of numbers. Over the past decade, the sampling theory has undergone a strong revival and the standard sampling paradigm is extended to non-bandlimited signals including signals in reproducing kernel spaces (RKSs), signals with finite rate of innovation (FRI) and sparse signals, and to nontraditional sampling methods, such as phaseless sampling. In this dissertation, we first consider the sampling and Galerkin reconstruction in a reproducing kernel space. The fidelity measure of perceptual signals, such as acoustic and visual signals, might not be well measured by least squares. In the first part of this dissertation, we introduce a fidelity measure depending on a given sampling scheme and propose a Galerkin method in Banach space setting for signal reconstruction. We show that the proposed Galerkin method provides a quasi-optimal approximation, and the corresponding Galerkin equations could be solved by an iterative approximation-projection algorithm in a reproducing kernel subspace of Lp. A spatially distributed network contains a large amount of agents with limited sensing, data processing, and communication capabilities. Recent technological advances have opened up possibilities to deploy spatially distributed networks for signal sampling and reconstruction. We introduce a graph structure for a distributed sampling and reconstruction system by coupling agents in a spatially distributed network with innovative positions of signals. We split a distributed sampling and reconstruction system into a family of overlapping smaller subsystems, and we show that the stability of the sensing matrix holds if and only if its quasi-restrictions to those subsystems have l_2 uniform stability. This new stability criterion could be pivotal for the design of a robust distributed sampling and reconstruction system against supplement, replacement and impairment of agents, as we only need to check the uniform stability of affected subsystems. We also propose an exponentially convergent distributed algorithm for signal reconstruction, that provides a suboptimal approximation to the original signal in the presence of bounded sampling noises. Phase retrieval (Phaseless Sampling and Reconstruction) arises in various fields of science and engineering. It consists of reconstructing a signal of interest from its magnitude measurements. Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. We consider phaseless sampling and reconstruction of real-valued signals in a shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. We find an equivalence between nonseparability of signals in a shift-invariant space and their phase retrievability with phaseless samples taken on the whole Euclidean space. We also introduce an undirected graph to a signal and use connectivity of the graph to characterize the nonseparability of high-dimensional signals. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that signals in shift-invariant spaces, that are determined by their magnitude measurements on the whole Euclidean space, can be reconstructed in a stable way from their phaseless samples taken on that discrete set. We also propose a reconstruction algorithm which provides a suboptimal approximation to the original signal when its noisy phaseless samples are available only

Sensitivity Analysis and Optimization of Aerodynamic Configurations with Blend Surfaces

A novel (geometrical) parametrization procedure using solutions to a suitably chosen fourth order partial differential equation is used to define a class of airplane configurations. Inclusive in this definition are surface grids, volume grids, and grid sensitivity. The general airplane configuration has wing, fuselage, vertical tail and horizontal tail. The design variables are incorporated into the boundary conditions, and the solution is expressed as a Fourier series. The fuselage has circular cross section, and the radius is an algebraic function of four design parameters and an independent computational variable. Volume grids are obtained through an application of the Control Point Form method. A graphic interface software is developed which dynamically changes the surface of the airplane configuration with the change in input design variable. The software is made user friendly and is targeted towards the initial conceptual development of any aerodynamic configurations. Grid sensitivity with respect to surface design parameters and aerodynamic sensitivity coefficients based on potential flow is obtained using an Automatic Differentiation precompiler software tool ADIFOR. Aerodynamic shape optimization of the complete aircraft with twenty four design variables is performed. Unstructured and structured volume grids and Euler solutions are obtained with standard software to demonstrate the feasibility of the new surface definition

Inference of Gaussian graphical models and ordinary differential equations

Netwerken vormen een handig instrument bij het visualiseren van systemen bestaand uit elementen die onderling interactie aangaan. Genregulatienetwerken, bijvoorbeeld, zijn complexe systemen die bestaan uit genen, eiwitten en andere moleculen. De elementen van een dergelijk systeem worden weergegeven door knooppunten, die door lijnen worden verbonden op het moment dat de bijbehorende elementen met elkaar in interactie zijn.In veel wetenschappelijke disciplines vormt het blootleggen van de structuur van een netwerk een belangrijk en ingewikkeld probleem. Vaak is er weinig bekend over een systeem en moet men uitgaan van meetgegevens uit knooppunten om een inschatting te kunnen maken van de structuur van het bijbehorende netwerk. Deze meetgegevens zijn echter aan ruis onderhevig. Wanneer de structuur van de interacties bekend is, staan we voor een andere uitdaging: of het nou gaat om het beschrijving van bruggen die een hevige wind moeten weerstaan of om de verspreiding van een infectieziekte, de vraag is hoe we op basis van dezelfde aan ruis onderhevige data kunnen bepalen hoe de fijne dynamica van het systeem in elkaar zit.In dit proefschrift stellen we enkele aanpassingen voor op bestaande methodes, om het schatten van de structuur van en interacties binnen netwerken en dynamische systemen te verbeteren.Enkele toepassingen van de methodes die we ontwikkelen zijn: het voorspellen van het aantal individuen dat tijdens de kindertijd mazelen krijgt, en inferentie van de interactie tussen genen en eiwitten in de E. colibacterie.Networks provide a simple way to visualize a system of interacting elements. For example, gene regulatory networks are complex systems whose elements are genes, proteins and other molecules. The elements of this system are represented by nodes and lines are drawn between them if they interact with each other. In many sciences uncovering the network structure is an important and difficult problem. With a limited knowledge about the system noisy measurements on the nodes should be used to estimate the network. When we know the structure of the interactions, another major obstacle is to learn the fine dynamics of the system using the same noisy data, from describing bridges subject to strong winds to the spread of an infectious disease.In this thesis we propose modifications of existing methods to improve the estimation of networks and dynamical systems.Some applications of methods we develop include: predicting the number of individuals that get infected by childhood disease measles, reconstructing transcription factor activities in streptomyces coelicolor bacterium, and inferring the interaction between genes and proteins in Escherichia coli bacterium