Unsteady melting of viscous material near stagnation point
Boundary value problem computer program for theoretical computation of unsteady melting of viscous glassy material near stagnation point
UNSTEADY FLOW↗
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Electrochemomechanics with flexoelectricity and modelling of electrochemical strain microscopy in mixed ionic-electronic conductors
Recently, a new scanning probe microscopy approach, referred to as electrochemical strain microscopy (ESM), for probing local ionic flows and electrochemical reactions in solids based on the bias-strain coupling was proposed by Morozovska et al. Then, a series of theoretical papers for analyzing the image formation and spectroscopic mechanism of ESM were published within the framework of Fermi-Dirac statistics, the Vegard law, the direct flexoelectric coupling effect, the electrostriction effect, and so on. However, most of the models in these papers are limited to the partial coupling or particular process, and numerically solved by using decoupling approximation. In this paper, to model the ESM measurement with the coupling electrical-chemical-mechanical process, the chemical Gibbs function variational principle for the thermal electrical chemical mechanical fully coupling problem is proposed. The fully coupling governing equations are derived from the variational principle. When the tip concentrates the electric field within a small volume of the material, the inhomogeneous electric field is induced. So, both direct and inverse flexoelectric effects should be taken into account. Here, the bulk defect electrochemical reactions are also taken into account, which are usually omitted in the existing works. This theory can be used to deal with coupling problems in solids, including conductors, semiconductors, and piezoelectric and non-piezoelectric dielectrics. As an application of this work, a developed initial-boundary value problem is solved numerically in a mixed ion-electronic conductor. Numerical results show that it is meaningful and necessary to consider the bulk defect chemical reaction. Besides, the chemical reaction and the flexoelectric effect have an interactive influence on each other. This work can provide theoretical basis for the ESM as well as investigating the bulk chemical reaction process in solids.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Study of Creep and Damage for a Hollow Cylinder on the Basis of Space and Refined Shell Models
We consider the axisymmetric problem of creep and creep damage of a hollow cylinder and develop a numerical-analytic method for the solution of this nonlinear initial-boundary-value problem. We also present an example of evaluation of creep and the time to fracture for a cylinder loaded by internal pressure. The results accumulated by using a three-dimensional model are compared with the data obtained on the bases of the refined model of the theory of shells.
97 MATHEMATICS AND COMPUTING↗
On Stable Wall Boundary Conditions for the Hermite Discretization of the Linearised Boltzmann Equation
We define certain criteria, using the characteristic decomposition of the boundary conditions and energy estimates, which a set of stable boundary conditions for a linear initial boundary value problem, involving a symmetric hyperbolic system, must satisfy. We first use these stability criteria to show the instability of the Maxwell boundary conditions proposed by Grad (Commun Pure Appl Math 2(4):331–407, 1949). We then recognise a special block structure of the moment equations which arises due to the recursion relations and the orthogonality of the Hermite polynomials; the block structure will help us in formulating stable boundary conditions for an arbitrary order Hermite discretization of the Boltzmann equation. The formulation of stable boundary conditions relies upon an Onsager matrix which will be constructed such that the newly proposed boundary conditions stay close to the Maxwell boundary conditions at least in the lower order moments.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Cutoff frequencies of eccentric waveguides
Boundary value problem solved by point matching method in study of circular eccentric waveguide cut-off frequencies
POINT MATCHING METHOD↗
Hermite analogs of the lowest order Raviart–Thomas mixed method for convection–diffusion equations
The Raviart–Thomas mixed finite-element method of the lowest order (Raviart and Thomas in mixed finite-element methods for second-order elliptic problems, Lecture Notes in mathematics, Springer, New York, 1977) commonly known as the RT{sub 0} method, is a well-established and popular numerical tool to solve diffusion-like problems providing flux continuity across inter-element boundaries. Douglas and Roberts extended the method to the case of more general second-order boundary-value problems including the convection–diffusion equations (cf. this journal Douglas in Comput Appl Math 1:91–103; 1982). The main drawback of these methods, however, is the poor representation of the primal variable by piecewise constant functions. The Hermite analog of the RT{sub 0} method for treating pure diffusion phenomena proposed in Ruas (J Comput Appl Math, 246:234–242; 2013) proved to be a valid alternative to attain higher order approximation of the primal variable while keeping intact the matrix structure and the quality of the discrete flux variable of the original RT{sub 0} method. Non-trivial extensions of this method are studied here that can be viewed as Hermite analogs of the two Douglas and Roberts’ versions of the RT{sub 0} method, to solve convection–diffusion equations. A detailed convergence study is carried out for one of the Hermite methods, and numerical results illustrate the performance of both of them, as compared to each other and to the corresponding mixed methods.
97 MATHEMATICS AND COMPUTING↗
On Stability of Perturbed Semigroups in Partially Ordered Banach Spaces
We prove necessary and sufficient stability conditions for perturbed semigroups of linear operators in Banach spaces with cones and consider examples using these conditions. In particular, we consider an example where the boundary-value problem is perturbed by a linear operator with a delayed independent variable and establish stability conditions for such a perturbed semigroup.
97 MATHEMATICS AND COMPUTING↗
Analytic in a Sector Resolving Families of Operators for Degenerate Evolution Fractional Equations
We introduce a class of pairs of operators defining a linear homogeneous degenerate evolution fractional differential equation in a Banach space. Reflexive Banach spaces are represented as the direct sums of the phase space of the equation and the kernel of the operator at the fractional derivative. In a sector of the complex plane containing the positive half-axis, we construct an analytic family of resolving operators that degenerate only on the kernel. The results are used in the study of the solvability of initial-boundary value problems for partial differential equations containing fractional time-derivatives and polynomials in the Laplace operator with respect to the spatial variable.
97 MATHEMATICS AND COMPUTING↗
Propagation of Boundary-Induced Discontinuity in Stationary Radiative Transfer
We consider the boundary value problem of the stationary transport equation in the slab domain of general dimensions. In this paper, we discuss the relation between discontinuity of the incoming boundary data and that of the solution to the stationary transport equation. We introduce two conditions posed on the boundary data so that discontinuity of the boundary data propagates along positive characteristic lines as that of the solution to the stationary transport equation. Our analysis does not depend on the celebrated velocity averaging lemma, which is different from previous works. We also introduce an example in two dimensional case which shows that piecewise continuity of the boundary data is not a sufficient condition for the main result.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
On New Structures in the Theory of Fully Nonlinear Equations
We describe the current state of the theory of equations with m-Hessian stationary and evolution operators. It is quite important that new algebraic and geometric notions appear in this theory. In the present work, a list of those notions is provided. Among them, the notion of m-positivity of matrices is quite important; we provide a proof of an analog of Sylvester’s criterion for such matrices. From this criterion, we easily obtain necessary and sufficient conditions for existence of classical solutions of the first initial boundary-value problem for m-Hessian evolution equations. The asymptotic behavior of m-Hessian evolutions in a semibounded cylinder is considered as well.
97 MATHEMATICS AND COMPUTING↗
Meso-Scale Approximations of Fields Around Clusters of Defects
We present a review of recent results on asymptotic approximations of solutions to boundary value problems in domains with large clusters of small defects. There are no assumptions which require periodicity within the cluster. The asymptotic approximations we discuss are uniform and include the boundary layers occurring in neighborhoods of singularly perturbed boundaries of the domains concerned. The term “meso-scale” is used to describe these approximations since they go beyond the conventional constraints of the homogenization theory.
97 MATHEMATICS AND COMPUTING↗
Enforcing global constraints for the dispersion closure problem: τ 2 -SIMPLE algorithm
Permeability and effective dispersion tensors are critical parameters to characterize flow and transport in porous media at the continuum scale. Homogenization theory defines a framework in which such effective properties are first computed from solving a closure problem in a repeating unit cell of the periodic microstructure and then used in a macroscopic formulation for efficient computation. The closure problem is formulated as a local boundary value problem subjected to global constraints, which guarantee the uniqueness of the solution and can be difficult to satisfy for complex geometries and at high flow conditions. These constraints also ensure that pore-scale pressure, velocity, and concentration fields can be accurately reconstructed from the closure variable. Building on a previous work, here we present a framework that allows to satisfy global constraints associated to both the permeability and the dispersion closure problems by introducing two artificial time scales. The algorithm, called τ 2 -SIMPLE, computes both permeability and effective dispersion given an arbitrarily complex geometry and flow condition. Furthermore, this algorithm is demonstrated to be accurate for both 2D and 3D geometries across varying flow conditions, and thus it can be used to quickly characterize effective properties from porous media images in many applications.
97 MATHEMATICS AND COMPUTING↗
Application of the inverse fast multipole method as a preconditioner in a 3D Helmholtz boundary element method
Here we investigate an efficient preconditioning of iterative methods (such as GMRES) for solving dense linear systems Ax = b that follow from a boundary element method (BEM) for the 3D Helmholtz equation, focusing on the low-frequency regime. While matrix–vector products in GMRES can be accelerated through the low-frequency fast multipole method (LFFMM), the BEM often remains computationally expensive due to the large number of GMRES iterations. We propose the application of the inverse fast multipole method (IFMM) as a preconditioner to accelerate the convergence of GMRES. The IFMM is in essence an approximate direct solver that uses a multilevel hierarchical decomposition and low-rank approximations. The proposed IFMM-based preconditioning has a tunable parameter ε that balances the cost to construct a preconditioner M, which is an approximation of A -1 , and the cost to perform the iterative process by means of M. Namely, using a small (respectively, large) value of ε takes a long (respectively, short) time to construct M, while the number of iterations can be small (respectively, large). A comprehensive set of numerical examples involving various boundary value problems with complicated geometries and mixed boundary conditions is presented to validate the efficiency of the proposed method. We show that the IFMM preconditioner (with a nearly optimal ε of 10 -2 ) clearly outperforms some common preconditioners for the BEM, achieving 1.2–10.8 times speed-up of the computations, in particular when the scale of the underlying scatterer is about five wavelengths or more. In addition, the IFMM preconditioner is capable of solving complicated problems (in a reasonable amount of time) that BD preconditioner can not.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Investigation of technical problems related to retrieval of uncooperative orbiting objects
Preliminary work on high altitude deployment of payloads was carried out. The problem of optimal transfer of the retrieval package from space shuttle to target was studied. The problem was characterized as a nonlinear boundary value problem with a terminal constraint on approach direction. Methods of identifying spin axis and direction are considered.
Kaplan, M. H.↗
Electromagnetic wave propagation in a magneto-plasma filled coaxial structure. I - Theoretical. II - Experimental
This study is concerned with the problem of electromagnetic wave propagation in a magneto-plasma filled coaxial structure. The problem is formulated using the classical boundary value problem approach. A numerical investigation shows the existence of propagating slow modes, backward modes, a quasi-TEM mode, and waveguide-type modes in a magneto-plasma filled coaxial structure. Dispersion curves for these different modes are presented. Measurements have been made of electromagnetic propagation in a coaxial electrode structure filled with longitudinally magnetized plasma. The annular plasma region had a 9.55 cm outer diameter, a 3.82 cm inner diameter and was approximately 60 cm long. A magnetic field of 300 gauss was employed. Electromagnetic wave frequencies were in the range .5 to 2.4 GHz. The plasma was generated by a continuous glow discharge. The resulting dispersion curves closely follow the predicted curves for the quasi-TEM mode.
Askins, H. W., Jr.↗
Numerical quadrature methods for integrals of singular periodic functions and their application to singular and weakly singular integral equations
High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.
Sidi, A.↗
Optimal Angle of Attack Control for Aerocapture Maneuvers
Recent investigations of aerocapture maneuvers have considered direct force control as a promising alternative to bank angle control. By modulating the angle of attack and sideslip angle independently, the longitudinal and lateral channels can be decoupled and thus solved separately. However, the optimal angle of attack profile for minimizing post-atmospheric propellant expenditure was previously unknown. This research applies principles of optimal control theory to numerically solve for the optimal angle of attack profile for control of the longitudinal channel. Direct methods are first employed to discretize the angle of attack profile and show that an optimal solution follows a nearly bang-bang (full lift up then full lift down) structure under a variety of conditions along the flight path angle corridor. To corroborate these results, indirect methods were also applied to scope the Optimal Control Problem (OCP) as a Two Point Boundary Value Problem (2P-BVP) which could then be solved numerically. The indirect optimization results aligned closely with the direct results, providing further evidence that a bang-bang structure provides a good approximation for the optimal angle of attack profile for aerocapture maneuvers using direct force control.
aerocapture↗
Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials
We construct energy-dependent potentials for which the Schrödinger equations admit solutions in terms of exceptional orthogonal polynomials. Our method of construction is based on certain point transformations, applied to the equations of exceptional Hermite, Jacobi and Laguerre polynomials. We present several examples of boundary-value problems with energy-dependent potentials that admit a discrete spectrum and the corresponding normalizable solutions in closed form.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗