Transients Of Platoons With Asymmetric And Different Laplacians, 2016 Czech Technical University, Prague

#### Transients Of Platoons With Asymmetric And Different Laplacians, Ivo Herman, Dan Martinec, J. J. P. Veerman

*J. J. P. Veerman*

We consider an asymmetric control of platoons of identical vehicles with nearest-neighbor interaction. Recent results show that if the vehicle uses different asymmetries for position and velocity errors, the platoon has a short transient and low overshoots. In this paper we investigate the properties of vehicles with friction. To achieve consensus, an integral part is added to the controller, making the vehicle a third-order system. We show that the parameters can be chosen so that the platoon behaves as a wave equation with different wave velocities. Simulations suggest that our system has a better performance than other nearest-neighbor scenarios. Moreover ...

A New Error Bound For Linear Complementarity Problems For B-Matrices, 2016 Yunnan University

#### A New Error Bound For Linear Complementarity Problems For B-Matrices, Chaoqian Li, Mengting Gan, Shaorong Yang

*Electronic Journal of Linear Algebra*

A new error bound for the linear complementarity problem is given when the involved matrix is a $B$-matrix. It is shown that this bound improves the corresponding result in [M. Garc\'{i}a-Esnaola and J.M. Pe\~{n}a. Error bounds for linear complementarity problems for $B$-matrices. {\em Appl. Math. Lett.}, 22:1071--1075, 2009.] in some cases, and that it is sharper than that in [C.Q. Li and Y.T. Li. Note on error bounds for linear complementarity problems for $B$-matrices. {\em Appl. Math. Lett.}, 57:108--113, 2016.].

Optimization Of Takeoffs On Unbalanced Fields Using Takeoff Performance Tool, 2016 AAR Aerospace Consulting, LLC

#### Optimization Of Takeoffs On Unbalanced Fields Using Takeoff Performance Tool, Nihad E. Daidzic

*International Journal of Aviation, Aeronautics, and Aerospace*

Unbalanced field length exists when ASDA and TODA are not equal. Airport authority may add less expensive substitutes to runway full-strength pavement in the form of stopways and/or clearways to basic TORA to increase operational takeoff weights. Here developed Takeoff Performance Tool is a physics-based total-energy model used to simulate FAR/CS 25 regulated airplane takeoffs. Any aircraft, runway, and environmental conditions can be simulated, while complying with the applicable regulations and maximizing performance takeoff weights. The mathematical model was translated into Matlab, Fortran 95/2003/2008, Basic, and MS Excel computer codes. All existing FAR/CS 25 takeoff ...

Signal Velocity In Oscillator Networks, 2016 Portland State University

#### Signal Velocity In Oscillator Networks, Carlos E. Cantos, J.J. P. Veerman, David K. Hammond

*J.J.P. Veerman*

We investigate a system of coupled oscillators on the circle, which arises from a simple model for behavior of large numbers of autonomous vehicles. The model considers asymmetric, linear, decentralized dynamics, where the acceleration of each vehicle depends on the relative positions and velocities between itself and a set of local neighbors. We first derive necessary and sufficient conditions for asymptotic stability, then derive expressions for the phase velocity of propagation of disturbances in velocity through this system. We show that the high frequencies exhibit damping, which implies existence of well-defined signal velocities c+>0 and c−f(x−c ...

Full State Revivals In Linearly Coupled Chains With Commensurate Eigenspectra, 2016 Portland State University

#### Full State Revivals In Linearly Coupled Chains With Commensurate Eigenspectra, J.J. P. Veerman, Jovan Petrovic

*J.J.P. Veerman*

Coherent state transfer is an important requirement in the construction of quantum computer hardware. The state transfer can be realized by linear next-neighbour-coupled finite chains. Starting from the commensurability of chain eigenvalues as the general condition of periodic dynamics, we find chains that support full periodic state revivals. For short chains, exact solutions are found analytically by solving the inverse eigenvalue problem to obtain the coupling coefficients between chain elements. We apply the solutions to design optical waveguide arrays and perform numerical simulations of light propagation thorough realistic waveguide structures. Applications of the presented method to the realization of a ...

Transients In The Synchronization Of Oscillator Arrays, 2016 Portland State University

#### Transients In The Synchronization Of Oscillator Arrays, Carlos E. Cantos, J.J. P. Veerman

*J.J.P. Veerman*

The purpose of this note is threefold. First we state a few conjectures that allow us to rigorously derive a theory which is asymptotic in N (the number of agents) that describes transients in large arrays of (identical) linear damped harmonic oscillators in R with completely decentralized nearest neighbor interaction. We then use the theory to establish that in a certain range of the parameters transients grow linearly in the number of agents (and faster outside that range). Finally, in the regime where this linear growth occurs we give the constant of proportionality as a function of the signal velocities ...

On Rank Driven Dynamical Systems, 2016 Portland State University

#### On Rank Driven Dynamical Systems, J.J. P. Veerman, F. J. Prieto

*J.J.P. Veerman*

We investigate a class of models related to the Bak-Sneppen model, initially proposed to study evolution. The BS model is extremely simple and yet captures some forms of “complex behavior” such as self-organized criticality that is often observed in physical and biological systems. In this model, random fitnesses in [0, 1] are associated to agents located at the vertices of a graph G. Their fitnesses are ranked from worst (0) to best (1). At every time-step the agent with the worst fitness and some others with a priori given rank probabilities are replaced by new agents with random fitnesses. We ...

Regularity Of Mediatrices In Surfaces, 2016 Portland State University

#### Regularity Of Mediatrices In Surfaces, Pilar Herreros, Mario Ponce, J.J. P. Veerman

*J.J.P. Veerman*

For distinct points p and q in a two-dimensional Riemannian manifold, one defines their mediatrix Lpq as the set of equidistant points to p and q. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. This paper establishes additional geometric regularity properties of mediatrices. We show that mediatrices have the radial linearizability property, which implies that at each point they have a geometrically defined derivative in the branching directions. Also, we study the particular case of mediatrices on spheres, by showing that they are Lipschitz simple closed curves ...

On A Convex Set With Nondifferentiable Metric Projection, 2016 Portland State University

#### On A Convex Set With Nondifferentiable Metric Projection, Shyan S. Akmal, Nguyen Mau Nam, J.J. P. Veerman

*J.J.P. Veerman*

A remarkable example of a nonempty closed convex set in the Euclidean plane for which the directional derivative of the metric projection mapping fails to exist was constructed by A. Shapiro. In this paper, we revisit and modify that construction to obtain a convex set with smooth boundary which possesses the same property.

Model For Computing Kinetics Of The Graphene Edge Epitaxial Growth On Copper, 2016 Western Kentucky University

#### Model For Computing Kinetics Of The Graphene Edge Epitaxial Growth On Copper, Mikhail Khenner

*Mikhail Khenner*

Dynamics Of Locally Coupled Oscillators With Next-Nearest-Neighbor Interaction, 2016 University of Crete

#### Dynamics Of Locally Coupled Oscillators With Next-Nearest-Neighbor Interaction, J. Herbrych, A. G. Chazirakis, N. Christakis, J. J. P. Veerman

*J.J.P. Veerman*

A theoretical description of decentralized dynamics within linearly coupled, one-dimensional oscillators (agents) with up to next-nearest-neighbor interaction is given. Conditions for stability of such system are presented. Our results indicate that the stable systems have response that grow at least linearly in the system size. We give criteria when this is the case. The dynamics of these systems can be described with traveling waves with strong damping in the high frequencies. Depending on the system parameters, two types of solutions have been found: damped oscillations and reflectionless waves. The latter is a novel result and a feature of systems with ...

Unified Hybridization Of Discontinuous Galerkin, Mixed, And Continuous Galerkin Methods For Second Order Elliptic Problems, 2016 University of Minnesota - Twin Cities

#### Unified Hybridization Of Discontinuous Galerkin, Mixed, And Continuous Galerkin Methods For Second Order Elliptic Problems, Bernardo Cockburn, Jay Gopalakrishnan, Raytcho Lazarov

*Jay Gopalakrishnan*

We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continu- ous Galerkin, non-conforming and a new, wide class of hybridizable discontinuous Galerkin methods. The distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements. Since the associated matrix is sparse, symmetric and positive definite, these methods can be efficiently implemented. Moreover, the ...

The Derivation Of Hybridizable Discontinuous Galerkin Methods For Stokes Flow, 2016 University of Minnesota - Twin Cities

#### The Derivation Of Hybridizable Discontinuous Galerkin Methods For Stokes Flow, Bernardo Cockburn, Jay Gopalakrishnan

*Jay Gopalakrishnan*

In this paper, we introduce a new class of discontinuous Galerkin methods for the Stokes equations. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries. We present four ways of hybridizing the methods, which differ by the choice of the globally coupled unknowns. Classical methods for the Stokes equations can be thought of as limiting cases of these new methods.

The Convergence Of V-Cycle Multigrid Algorithms For Axisymmetric Laplace And Maxwell Equations, 2016 Portland State University

#### The Convergence Of V-Cycle Multigrid Algorithms For Axisymmetric Laplace And Maxwell Equations, Jay Gopalakrishnan, Joseph E. Pasciak

*Jay Gopalakrishnan*

We investigate some simple finite element discretizations for the axisymmetric Laplace equation and the azimuthal component of the axisymmetric Maxwell equations as well as multigrid algorithms for these discretizations. Our analysis is targeted at simple model problems and our main result is that the standard V-cycle with point smoothing converges at a rate independent of the number of unknowns. This is contrary to suggestions in the existing literature that line relaxations and semicoarsening are needed in multigrid algorithms to overcome difficulties caused by the singularities in the axisymmetric Maxwell problems. Our multigrid analysis proceeds by applying the well known regularity ...

Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, 2016 Portland State University

#### Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, Jay Gopalakrishnan, Joseph E. Pasciak

*Jay Gopalakrishnan*

The finite element method when applied to a second order partial differential equation in divergence form can generate operators that are neither Hermitian nor definite when the coefficient function is complex valued. For such problems, under a uniqueness assumption, we prove the continuous dependence of the exact solution and its finite element approximations on data provided that the coefficients are smooth and uniformly bounded away from zero. Then we show that a multigrid algorithm converges once the coarse mesh size is smaller than some fixed number, providing an efficient solver for computing discrete approximations. Numerical experiments, while confirming the theory ...

Multigrid In A Weighted Space Arising From Axisymmetric Electromagnetics, 2016 Portland State University

#### Multigrid In A Weighted Space Arising From Axisymmetric Electromagnetics, Dylan M. Copeland, Jay Gopalakrishnan, Minah Oh

*Jay Gopalakrishnan*

Consider the space of two-dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable. This space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates. We prove that if the original three-dimensional domain is convex then the multigrid Vcycle applied to the inner product in this space converges, provided certain modern smoothers are used. For the convergence analysis, we first prove several intermediate results, e.g., the approximation properties of a commuting projector in weighted norms, and a superconvergence estimate for ...

Multigrid For The Mortar Finite Element Method, 2016 Portland State University

#### Multigrid For The Mortar Finite Element Method, Jay Gopalakrishnan, Joseph E. Pasciak

*Jay Gopalakrishnan*

A multigrid technique for uniformly preconditioning linear systems arising from a mortar finite element discretization of second order elliptic boundary value problems is described and analyzed. These problems are posed on domains partitioned into subdomains, each of which is independently triangulated in a multilevel fashion. The multilevel mortar finite element spaces based on such triangulations (which need not align across subdomain interfaces) are in general not nested. Suitable grid transfer operators and smoothers are developed which lead to a variable Vcycle preconditioner resulting in a uniformly preconditioned algebraic system. Computational results illustrating the theory are also presented.

Nédélec Spaces In Affine Coordinates, 2016 Portland State University

#### Nédélec Spaces In Affine Coordinates, Jay Gopalakrishnan, Luis E. García-Castillo, Leszek Demkowicz

*Jay Gopalakrishnan*

In this note, we provide a conveniently implementable basis for simplicial Nédélec spaces of any order in any space dimension. The main feature of the basis is that it is expressed solely in terms of the barycentric coordinates of the simplex.

Symmetric Nonconforming Mixed Finite Elements For Linear Elasticity, 2016 Portland State University

#### Symmetric Nonconforming Mixed Finite Elements For Linear Elasticity, Jay Gopalakrishnan, Johnny Guzmán

*Jay Gopalakrishnan*

We present a family of mixed methods for linear elasticity that yield exactly symmetric, but only weakly conforming, stress approximations. The method is presented in both two and three dimensions (on triangular and tetrahedral meshes). The method is efficiently implementable by hybridization. The degrees of freedom of the Lagrange multipliers, which approximate the displacements at the faces, solve a symmetric positive-definite system. The design and analysis of this method is motivated by a new set of unisolvent degrees of freedom for symmetric polynomial matrices. These new degrees of freedom are also used to give a new simple calculation of the ...

Polynomial Extension Operators. Part I, 2016 University of Texas at Austin

#### Polynomial Extension Operators. Part I, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

*Jay Gopalakrishnan*

In this series of papers, we construct operators that extend certain given functions on the boundary of a tetrahedron into the interior of the tetrahedron, with continuity properties in appropriate Sobolev norms. These extensions are novel in that they have certain polynomial preservation properties important in the analysis of high order finite elements. This part of the series is devoted to introducing our new technique for constructing the extensions, and its application to the case of polynomial extensions from H^{½}(∂K) into H¹(K), for any tetrahedron K.