Unbounded Region Research Articles

Location of eigenvalues of three-dimensional non-self-adjoint Dirac operators

We prove the absence of eigenvaues of the three-dimensional Dirac operator with non-Hermitian potentials in unbounded regions of the complex plane under smallness conditions on the potentials in Lebesgue spaces. Our sufficient conditions are quantitative and easily checkable.

Handling bounded and unbounded unsafe sets in Control Lyapunov-Barrier function-based model predictive control of nonlinear processes

Control Lyapunov-Barrier function (CLBF) has been used to design controllers for nonlinear systems subject to input constraints to ensure closed-loop stability and process operational safety simultaneously. In this work, we developed Control Lyapunov-Barrier functions for two types of unsafe regions (i.e., bounded and unbounded sets) to solve the problem of stabilization of nonlinear systems with guaranteed process operational safety. Specifically, in the presence of a bounded unsafe region embedded within the closed-loop system stability region, the CLBF-MPC is developed by incorporating CLBF-based constraints and discontinuous control actions at potential stationary points (except the origin) to guarantee the convergence to the origin (i.e., closed-loop stability) and the avoidance of unsafe region (i.e., process operational safety). In the case of unbounded unsafe sets, closed-loop stability with safety is readily guaranteed under the CLBF-MPC since the origin is the unique stationary point in state-space. The application of the proposed CLBF-MPC method is demonstrated through a chemical process example with a bounded and an unbounded unsafe region, respectively.

On the Coupling of VEM and BEM in Two and Three Dimensions

This paper introduces and analyzes the combined use of the virtual element method (VEM) and the boundary element method (BEM) to numerically solve linear transmission problems in two and three dimensions. As a model, we consider an elliptic equation in divergence form holding in an annular domain coupled with the Laplace equation in the corresponding unbounded exterior region, together with transmission conditions on the interface and a suitable radiation condition at infinity. We employ the usual primal formulation in the bounded region, and combine it, by means of the Costabel and Han approach, with the boundary integral equation method in the exterior domain. As a consequence, and in addition to the original unknown of the model, its normal derivative in two dimensions, and both its normal derivative and its trace in the three-dimensional case, are introduced as auxiliary nonvirtual unknowns. Moreover, for the latter case, a new and more suitable variational formulation for the coupling is introduced. In turn, the main ingredients required by the discrete analyses include the virtual element subspaces for the domain unknowns, explicit polynomial subspaces for the boundary unknowns, and suitable projection and interpolation operators that allow us to define the corresponding discrete bilinear forms. Then, as for the continuous formulations, the classical Lax--Milgram lemma is employed to derive the well-posedness of our coupled VEM/BEM scheme. A priori error estimates in the energy and weaker norms, and corresponding rates of convergence for the solution as well as for a fully computable projection of its virtual component, are provided. Finally, a couple of numerical examples in two dimensions illustrating the performance of the derived VEM/BEM schemes are reported.

Optimization of Edge Termination Techniques for β-Ga2O3 Schottky Rectifiers

To realize the potential of Gallium oxide (Ga2O3) Schottky rectifiers fabricated for high voltage and fast switching applications, various edge termination techniques to maximize the breakdown voltage (Vbr) are studied and examined via simulations using the FLOODS/FLOOPS TCAD simulator. The simulated Schottky rectifiers consist of a Si-doped (n = 1.0 × 1015 – 1.3 × 1017 cm−3) β-Ga2O3 epitaxial layer grown on Sn-doped (n = 4.8 × 1018 cm−3) Ga2O3 substrates. The optimization of field plate geometry for Schottky barrier diodes (SBD) was investigated using the device breakdown characteristics as the figure-of-merit. Various field plate dielectrics (SiO2, SiNx, Al2O3, and HfO2) were explored while the field plate structure was concurrently varied to obtain a normalized breakdown field (VNbr) of ∼3 for a step (graduated form) dielectric with Al2O3 as the dielectric. Edge termination via the formation of resistive areas at the anode contact periphery via ion (argon) implantation was also examined for the SBDs since other edge termination techniques are ineffective due to lack of p-type doping in Ga2O3. The configuration of the implanted region was investigated and a VNbr of over 5 was achieved for diodes with an unbounded resistive region and an implantation depth of 50–100 nm.

Structure peculiarities of binary state diagrams of vanadium with neighboring elements of Mendeleev’s periodic table

Data on search for correlations between a structure of state diagrams of binary systems based on V-Me (Me = Ti, Zr, Hf, Nb, Ta, Cr, Mo, W) and crystal-geometric and crystal-chemical parameters were presented. Manifestation of both positive and negative deviations in experimental concentration dependences of the atomic volume in solid solutions in systems V-Me (Me = Ti, Nb, Ta, Cr, Mo) from Zen’s law was established. It was found that in the systems with unbounded regions of solid solutions, the deviation from Zen’s law is insignificant. A correlation was found between the types of state diagrams and the difference in atomic volumes of alloy-forming elements in systems V-Me (Me = Ti, Nb, Ta, Cr, Mo).

Finite and infinite block Petrov–Galerkin method for cracks in functionally graded materials

• A novel finite and infinite block Petrov–Galerkin method is applied for crack problems in functionally graded materials . • The problem domain is discretized by a few blocks in the computation which saves tedious mesh generation . • The displacements and stresses are continuous at interfaces of neighboring blocks. • Due to the usage of Petrov–Galerkin formulation, the BPGM is a weak-form method. • Numerical results show that the proposed approach is accurate and efficient. This paper presents a novel weak-form block Petrov–Galerkin method (BPGM) for linear elastic and crack problems in functionally graded materials with bounded and unbounded problem domains. The main idea of this approach is to combine the meshless local Petrov–Galerkin method with block method. Once the problem domain is discretized into several sub-regions, named blocks, which can be mapped into normalized square domains. The weak-form Petrov–Galerkin method and polynomial series of interpolations are employed in each block. The computational efficiency is rigorously examined against the strong-form finite block method, the finite element technique and meshless approaches. Numerical results demonstrate that the BPGM possesses the following important properties: (1) only a few blocks are required for calculating problems in unbounded regions which saves tedious work of meshing; (2) the displacements and stresses are continuous at the interfaces of neighboring blocks; (3) due to the use of weak formulation, the continuity requirements of the approximation functions are reduced and numerical results are stable; (4) because of using Lagrange polynomial interpolation, highly accurate solutions can be obtained with a small amount of nodes in each block.

STUDI NAMA GEOGRAFI MELALUI LAYANAN PEMETAAN URUNDAYA DI DESA GIRIPURWO, PURWOSARI, GUNUNGKIDUL D.I. YOGYAKARTA

A geographical name is a name that identify specific feature on the earth. That features could be a settlement, administrative region, natural feature, artificial feature, unbounded region, or virtual region. Under the Law Number 4 of 2011 concerning Geospatial Information, the geographical name is one of the layer that must appear on the base map. The acquisition of geographical names can be facilitated by crowdsourcing map that are conducted by corporations or the public. The objectives of this study are 1) to carry out an inventory of geographic names through crowdsourced maps, and 2) to examine the opportunities and challenges of the study of geographic names in rural areas. We observed data from crowdsourcing maps, e.g., Google Maps, Here Maps, and OpenStreetMaps that cover Giripurwo Village. We used spatial comparison in this research. We also compared its appearances on various mapping scales. A field survey was conducted to get more qualitative information about geographical names and to test the accuracy of maps. The results showed that there were differences between the crowdsource map services in presenting the geographical names at the same scale level. We face constraints in this mapping, i.e. limited accessibility in the entire region and sparsely populated in a karst region. Conversely, the high participation of rural communities is beneficial in this mapping process.

Stability analysis of a multi-layer tumor model with free boundary

In this paper we study a multi-layer tumor model which is expressed as a free boundary problem of a system of partial differential equations. The problem consists of two elliptic equations describing the distribution of nutrient concentration and the pressure between tumor cells, respectively, in an unbounded strip-like region in which the tumor occupies. This region has two disjoint boundaries: While the lower part is fixed, the upper part, which stands for the tumor surface, can move as the tumor grows. Under certain conditions, the problem can be proved to admit a unique equilibrium which corresponds to the flat upper boundary. We first convert the model into a parabolic differential equation in certain function space. Next we compute the spectrum of the linearized problem at the equilibrium. By applying the geometric theory of parabolic differential equations in Banach spaces, we prove that if the cell-to-cell adhesiveness coefficient γ is larger than a threshold value γ∗, then the unique flat equilibrium is asymptotically stable, whereas in the case 0<γ<γ∗ the flat equilibrium is unstable.

On computing highly robust efficient solutions

AbstractDue to the real‐life issue of decision making in the presence of uncertainty and multiple conflicting objectives, it is of interest to solve optimization problems that incorporate these two aspects. To address this type of problem, we undertake the computation of highly robust efficient solutions to uncertain multiobjective linear programs with objective‐wise uncertainty in the cost matrix coefficients, and we address unbounded feasible regions as well as unbounded uncertainty sets. We provide methods for checking whether or not the highly robust efficient set is empty, computing highly robust efficient points, and determining whether a given solution of interest is highly robust efficient. An application in the area of bank management is included.

Dynamic response of a functionally graded tube embedded in an elastic medium due to SH-Waves

Dynamic response of a cylindrical tube surrounded by an unbounded elastic medium due to plane harmonic SH-Waves is studied. A two-dimensional mathematical model is considered. Cylindrical coordinates are used for convenience. The surrounding medium is assumed to be homogeneous, isotropic and linear elastic. The tube is assumed to be made of linear elastic functionally graded materials (FGMs) such that shear modulus and shear wave velocity are assumed to change linearly from inner surface to outer surface. Material properties are constant along circumferential direction. It is assumed that the inner surface of the tube is traction-free and there is a welded contact between the tube and the surrounding medium. Governing equations are slightly different in the tube region and the unbounded region. Both of the governing equations are solved by applying Finite Fourier Transform in circumferential direction. The exact solution series are presented in terms of Fourier-Bessel series in the unbounded region and power series in the tube region. The presented numerical results show that when the incoming wave lengths decrease, shear stresses at the tube increase significantly. It was shown that for the shorter incoming wave lengths, tubes made of FGMs are subjected to smaller shear stresses compared to the tubes homogeneously made of outer surface material of the FG cases.

Bernstein–Walsh type inequalities in unbounded regions with piecewise asymptotically conformal curve in the weighted Lebesgue space

We have obtained the pointwise Bernstein–Walsh type estimation for algebraic polynomials in the unbounded regions with piecewise asymptotically conformal boundary, having exterior and interior zero angles, in the weighted Lebesgue space.

On the numerical solution of the exterior elastodynamic problem by a boundary integral equation method

A numerical method for the Dirichlet initial boundary value problem for the elastic equation in the exterior and unbounded region of a smooth closed simply connected 2-dimensional domain, is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and a boundary integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the time-depended problem to a sequence of stationary boundary value problems, which are solved by a boundary layer approach resulting to a sequence of boundary integral equations of the first kind. The numerical discretization and solution are obtained by a trigonometrical quadrature method. Numerical results are included.

Modeling of shallow water waves with variable bathymetry in an irregular domain by using hybrid finite element method

An efficient numerical model is constructed with the combination of analytical and Hybrid Finite Element Method (HFEM) based on mild slope equation for shallow water waves to analyze the wave induced oscillation in an irregular geometrical domain. The domain of interest is divided in two regions as bounded and unbounded region. The solution of the mild slope equation is obtained in the bounded region with the consideration of variable bathymetry and partially reflected boundary using Hybrid element method. Further, an analytical approach is utilized in an unbounded region based on Fourier bessel series solution of scattered waves. Hybrid mesh elements is considered for bounded region to enhance the numerical accuracy. The numerical validation is conducted by the comparison of simulation results with analytical approximations and convergence analysis for the rectangular domain is also obtained. The amplification factor is obtained for T-shaped and TT-shaped domain to analyze the resonance modes. Finally, the current numerical model is applied on a realistic Pohang New Harbor (PNH) under the resonance conditions. Thus, the current numerical model can be utilized efficiently for redesigning and constructing artificial ports or harbors in the coastal regions with variable bathymetries.

Convective Wave Equation and Time Reversal Process for Source Refocusing

The convective wave equation deals with wave propagation in a moving media. We focus on the underwater acoustic wave equation where the convective element is the flow of water inside a river, along its length. The main thrust of this paper is the ill-posed “refocusing” problem. The initial condition simulates an explosion in a small compact region and the response is recorded over time at several microphones. Having only partial and noisy information we expect that small perturbations will destroy the ability to recover the complete initial data. We use the time reversal (TR) technique to determine the location of the original explosion, given limited spatial observations. We test the effectiveness of this scheme under conditions including dissipation, dispersion, etc. We use finite differences and implement absorbing boundary conditions to simulate an unbounded region.

Surface Green’s functions in finite plane elastostatics of harmonic materials

The closed-form representations of surface Green’s functions corresponding to the action of a concentrated force applied at the boundary of a region occupied by a particular class of compressible hyperelastic materials of harmonic type, has been derived. In our analysis, we consider both a bounded region in the form of a circular disk and an unbounded region with either an elliptical hole or a parabolic boundary.

Uniform and Pointwise Polynomial Estimates in Regions with Interior and Exterior Cusps

In&#x0D; this work, we investigate the estimation for algebraic polynomials in the&#x0D; bounded and unbounded regions with piecewise Dini smooth curve having interior&#x0D; and exterior zero angles.

Low-speed preconditioning for strongly coupled integration of Reynolds-averaged Navier–Stokes equations and two-equation turbulence models

Computational fluid dynamics codes using the density-based compressible flow formulation of the Navier–Stokes equations have proven to be very successful for the analysis of high-speed flows. However, solution accuracy degradation and, for explicit solvers, reduction of the residual convergence rates occur as the local Mach number decreases below the threshold of 0.1. This performance impairment worsens remarkably in the presence of flow reversals at wall boundaries and unbounded high-vorticity flow regions. These issues can be resolved using low-speed preconditioning, but there exists an outstanding problem regarding the use of this technology in the strongly coupled integration of the Reynolds-averaged Navier–Stokes equations and two-equation turbulence models, such as the k−ω shear stress transport model. It is not possible to precondition only the RANS equations without altering parts of the governing equations, and there did not exist an approach for preconditioning both the RANS and the SST equations. This study solves this problem by introducing a turbulent low-speed preconditioner of the RANS and SST equations that does not require any alteration of the governing equations. The approach has recently been shown to significantly improve convergence rates in the case of a one-equation turbulence model. The study focuses on the explicit multigrid integration of the governing equations, but most algorithms are applicable also to implicit integration methods. The paper provides all algorithms required for implementing the presented turbulent preconditioner in other computational fluid dynamics codes. The new method is applicable to all low- and mixed-speed aeronautical and propulsion flow problems, and is demonstrated by analyzing the flow field of a Darrieus wind turbine rotor section at two operating conditions, one of which is characterized by significant blade/vortex interaction. Verification and further validation of the new method is also based on the comparison of the results obtained with the developed density-based code and those obtained with a commercial pressure-based code.

Multiagent Rendezvous With Shortest Distance to Convex Regions With Empty Intersection: Algorithms and Experiments.

This paper presents both algorithms and experimental results to solve a distributed rendezvous problem with shortest distance to convex regions. In a multiagent network, each agent is assigned to a certain convex region and has information about only its own region. All these regions might not have an intersection. Through local interaction with their neighbors, multiple agents collectively rendezvous at an optimal location that is a priori unknown to each agent and has the shortest total squared distance to these regions. First, a distributed time-varying algorithm is introduced, where a corresponding condition is given to guarantee that all agents rendezvous at the optimal location asymptotically for bounded convex regions. Then a distributed tracking algorithm combined with a distributed estimation algorithm is proposed. It is first shown that for general possibly unbounded convex regions, all agents rendezvous in finite time and then collectively slide to the optimal location asymptotically. Then it is shown that for convex regions with certain projection compressibility, all agents collectively rendezvous at the optimal location in finite time, even when the regions are time varying. The algorithms are experimentally implemented on multiple ground robots to illustrate the obtained theoretical results.

Dynamics on the Double Morse Potential: A Paradigm for Roaming Reactions with no Saddle Points

In this paper we analyze a two degree of freedom Hamiltonian system constructed from two planar Morse potentials. The resulting potential energy surface has two potential wells surrounded by an unbounded flat region containing no critical points. In addition, the model has an index one saddle between the potential wells. We study the dynamical mechanisms underlying transport between the two potential wells, with emphasis on the role of the flat region surrounding the wells. The model allows us to probe many of the features of the roaming mechanism whose reaction dynamics are of current interest in the chemistry community.

The Schwartz Alternating Algorithm for Solving 3D Exterior Helmholtz Problems

Based on the natural boundary reduction, an overlapping domain decomposition method is discussed for solving exterior Helmholtz problem over a three-dimensional (3D) domain. By introducing two different artificial boundaries, the original unbounded domain is divided into a bounded subdomain and a typical unbounded region, and a Schwartz alternating method is presented. The finite element method and natural boundary element method are alternately applied to solve the problems in the bounded subdomain and the typical unbounded subdomain. Moreover, the convergence of the Schwartz alternating algorithm is studied. Finally, some numerical experiments are presented to show the performance of this method.