Two-dimensional Boundary Problem Research Articles

Multinode Shepard method for two-dimensional elliptic boundary problems on different shaped domains

In this paper, we continue the study on the application of multinode Shepard method to numerically solve elliptic Partial Differential Equations (PDEs) equipped with various conditions at the boundary of domains of different shapes. In particular, for the first time, the multinode Shepard method is proposed to solve elliptic PDEs with Dirichlet and/or Neumann boundary conditions. The method has been opportunely handled to efficiently work dealing with scattered distribution of points and, to this aim, several experiments in different 2d domains have been performed. Comparisons with the analytic solution and the results generated by the Kansa’s RBF solvers have been reported referring to Halton points. The results are very promising and should be of interest for applications in the real world.

Solving 2D boundary-value problems using discrete partial differential operators

PurposeDiscrete differential operators of periodic base functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary linear and nonlinear differential equations with Dirichlet and Neumann boundary conditions.Design/methodology/approachThis paper presents a promising approach for solving two-dimensional (2D) boundary problems of elliptic differential equations. To create finite differential equations, specially developed discrete partial differential operators are used to replace the partial derivatives in the differential equations. These operators relate the value of the partial derivatives at each point to the value of the function at all points evenly distributed over the area where the solution is being sought. Exemplary 2D elliptic equations are solved for two types of boundary conditions: the Dirichlet and the Neumann.FindingsAn alternative method has been proposed to create finite-difference equations and an effective method to determine the leakage flux in the transformer window.Research limitations/implicationsThe proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This method can be extended to the 3D or time-periodic 2D cases.Practical implicationsThis paper presents a methodology for calculations of the self- and mutual-leakage inductances for windings arbitrarily located in the transformer window, which is needed for special transformers or in any case of the internal asymmetry of windings.Originality/valueThe presented methodology allows us to obtain the magnetic vector potential distribution in the transformer window only, for example, to omit the magnetic core of the transformer from calculations.

Diffraction of flexural-gravity waves by a vertical cylinder of non-circular cross section

The linear three-dimensional problem of flexural-gravity wave (hydro-elastic wave) diffraction by a vertical cylinder of an arbitrary smooth cross section is studied using an asymptotic approach combined with the vertical mode method for water of finite depth. The surface of the water is covered by an infinite, continuous elastic ice plate. The rigid cylinder extends from the sea bottom to the ice surface. The ice plate is frozen to the cylinder. The ice deflection is described by the equation of a thin elastic plate of constant thickness with clamped edge conditions at the cylinder. The flow under the ice is described by the linear theory of potential flows. The coupled problem of wave diffraction is solved in two steps. First, the problem is solved without evanescent waves similar to the problem of water waves diffracted by a vertical cylinder. This solution does not satisfy the edge conditions. Second, a radiation problem with a prescribed motion of the ice plate edge is solved by the vertical mode method. The sum of these two solutions solve the original problem. Both solutions are obtained by an asymptotic method with a small parameter quantifying a small deviation of the cylinder cross section from a circular one. Third-order asymptotic solutions are obtained by solving a set of two-dimensional boundary problems for Helmholtz equations in the exterior of a circle. Strains along the edge, where the ice plate is frozen to the cylinder, are investigated for nearly square and elliptic cross sections of the vertical cylinders depending on the characteristics of ice and incident wave. The strains are shown to be highest in the places of high curvatures of the cross sections. The derived asymptotic formulae can be used in design of vertical columns in ice. They directly relate the strains in ice plate to the shape of the column.

Comparative Analysis of Plasma Equilibrium Computations in Toroidal and Cylindrical Magnetic Traps

Toroidal magnetic traps for plasma confinement make up an extended object of controlled nuclear fusion investigations. Mathematical simulation of equilibrium plasma configurations in the traps often deals with their analogues straightened into a cylinder. This paper presents a comparative analysis of their numerical investigations in both geometry variants. Mathematical tool of the models use two-dimensional boundary problems with the Grad-Shafranov differential equation for the magnetic flux function. As the investigation result, we present some quantitative characteristics of differences between toroidal and cylindrical configurations by two examples: a plasma torus with longitudinal electrical current and the Galathea-Belt toroidal trap with two ring-shaped current-carrying conductors immersed into the plasma.

TWO-STAGE GRID METHOD OF SOLUTION OF BOUNDARY PROBLEMS OF STRUCTURAL MECHANICS WITH THE USE OF DISCRETE HAAR BASIS

The distinctive paper is devoted to development of two-stage numerical method. At the first stage, the discrete problem is solved on a coarse grid, where the number of nodes in each direction is the same and is a pow-er of 2. Then the number of nodes in each direction is doubled and the resulting solution on a coarse grid using a discrete Haar basis is defined at the nodes of the fine grid as the initial approximation. At the second stage, we ob-tain a solution in the nodes of the fine grid using the most appropriate iterative method,. Test examples of the solu-tion of one-dimensional, two-dimensional and three-dimensional boundary problems are under consideration

Algorithm to construct bifurcation structure of non-linear boundary problem for von Karman equations

In the frameworks of the generalized Kantorovich method, a novel approach to detect and analyze singular points of a non-linear boundary problem for von Karman equations is proposed: an algorithm suggests that a sequence of single-dimensional boundary problems is constructed in order to solve the two-dimensional boundary problem in question. The aforesaid single-dimensional boundary problems are reduced to the equivalent Cauchy problems. In doing so, one calculates the Frechet matrix, whose degeneracy is necessary and sufficient conditions of branching. The simulation reveals the bifurcation structure for von Karman equations with the constant right term. In that case, the structure includes primary and secondary bifurcation paths.

Asymptotic analysis of solution to the nonlinear problem of non-stationary heat conductivity of layered anisotropic non-uniform shells at low Biot numbers on the front surfaces

The nonlinear problem of non-stationary heat conductivity of the layered anisotropic heat-sensitive shells was formulated taking into account the linear dependence of thermal-physical characteristics of the materials of phase compositions on the temperature. The initial-boundary-value problem is formulated in the dimensionless form, and four small parameters are identified: thermal-physical, characterizing the degree of heat sensitivity of the layer material; geometric, characterizing the relative thickness of the thin-walled structure, and two small Biot numbers on the front surfaces of shells. A sequential recursion of dimensionless equations is carried out, at first, using the thermalphysical small parameter, then, small Biot numbers and, finally, geometrical small parameter. The first type of recursion allowed us to linearize the problem of heat conductivity, and on the basis of two latter types of recursion, the outer asymptotic expansion of solution to the problem of non-stationary heat conductivity of the layered anisotropic non-uniform shells and plates under boundary conditions of the II and III kind and small Biot numbers on the facial surfaces was built, taking into account heat sensitivity of the layer materials. The resulting two-dimensional boundary problems were analyzed, and asymptotic properties of solutions to the heat conductivity problem were studied. The physical explanation was given to some aspects of asymptotic temperature decomposition.

A numerical solution to an inverse unsteady-state heat transfer problem involving the Trefftz functions

This paper shows the results concerning flow boiling heat transfer in an asymmetrically heated vertical minichannel. The heated element for FC-72 Fluorinert flowing in that minichannel was a thin foil. The foil surface temperature was monitored continuously at 18 points by K-type thermocouples from the outer foil surface. Fluid temperature and pressure in the minichannel inlet and outlet, current supplied to the foil and voltage drop were also monitored. Measurements were carried out at 1 s intervals. The objective was to determine the heat transfer coefficient on the heated foil–fluid contact surface in the minichannel. It was obtained from the Robin boundary condition. The foil temperature was the result of solving the nonstationary two-dimensional inverse boundary problem in the heated foil. Using the FEM combined with Trefftz functions as basis functions solved the problem. The unknown temperature values at nodes were calculated by minimising the adequate functional. The values of local heat transfer coefficients were consistent with the results obtained by the authors in their previous studies when steady-state conditions were analysed. This time, however, these values were analysed as time dependent, which facilitated observation of coefficient changes that were impossible to observe under the steady-state conditions.

Trefftz function-based thermal solution of inverse problem in unsteady-state flow boiling heat transfer in a minichannel

The paper shows the results for flow boiling heat transfer in a 1.7mm deep minichannel vertically oriented with FC-72 Fluorinert as a working fluid. The heated wall of the minichannel was formed with a thin foil. Thermocouples mounted at 18 points monitored the outer foil surface temperature. All experimental parameters were controlled using data acquisition stations. The measurements were performed at 0.01s intervals. The observations of the flow structures were carried out concurrently on the internal surface of the foil contacting the fluid. The aim of the numerical calculations was to determine the heat transfer coefficient on the contact surface between the heated foil and FC-72. The heat transfer coefficient was calculated with the use of the Robin boundary condition. To do that, the foil and fluid temperatures and foil temperature gradient had to be known. The foil temperature was found by solving an unsteady-state two-dimensional inverse boundary problem with the use of the Trefftz method in time-space subdomains. A linear combination of Trefftz functions was used to approximate the foil temperature. The unknown coefficients of the Trefftz function linear combination were determined by minimizing the functional. Error propagation in time and mean relative differences determined between the measured and computed heated foil temperatures were presented.

Scattering of light waves by finite metal nanostrip gratings: Nystrom-Type method and resonance effects

Efficient and rapidly convergent numerical algorithm for the simulation of the scattering of light waves by a finite gratings consisting of thin (thinner than the wavelength in the free space) metal nanostrips is presented. The model is based on the utilization of generalized boundary conditions (GBC), which allow one to exclude from consideration the field inside each strip and to reduce the two-dimensional boundary problem to one-dimensional systems of singular/hypersingular integral equations (IE). The obtained IE are solved numerically using the Nystrom-type method and the quadrature formulas of interpolation type, that provides guarantee convergence and controlled accuracy. The article presents the results of characteristics calculations for optical scattering and absorption by the gratings, which consist of silver nanostrips, as dependences on the width and on the thickness of the strips, and on the grating period. The nature of resonance phenomena has been investigated, namely the article presents the analysis of intensive optical scaterring and absorption in the case of excitation of plasmonic modes (plasmons) and of grating modes, which are induced by the periodicity.

Mathematical modeling of biological tissue cryodestruction

New statements of one- and two-dimensional Stefan-type boundary problems with nonlinear heat sources arising in cryosurgery are considered in the paper. An effective method for their full research for cases of sufficiently extended cryoinstruments with a cylindrical shape of a cooling surface is offered. Some results of the computer simulation are present.

Two-dimensional boundary problems of Stefan’s type in cryomedicine

In this paper we propose a new formulation of two-dimensional boundary problems of Stefan’s type with non-linear heat source of a special kind, arising in cryomedicine. These models correspond to the processes of biological tissue cooling by cryoinstruments with cylindrical and conical forms of the cooling surface. An effective method for the study of these models is offered.

Current-Phase Relation in Josephson Junctions with Complex Ferromagnetic/Normal Metal Interlayers

In the frame of Usadel equation the two-dimensional nonlinear boundary problem for heterostructures with complex ferromagnetic (F)/normal metal (N) interlayer has been solved numerically for arbitrary values of electrodes spacing L, thickness dN and dF of N and F films. We calculate the current-phase relations for ramp-type S-NF-S and overlap-type SFN-FN-NFS junctions. We have found the range of parameters providing an opportunity for realization of Josephson φ-junction based on the considered structures.

Плоские задачи со смешанными краевыми условиями для составной плоскости с трещинами.

Плоские задачи со смешанными краевыми условиями для составной плоскости с трещинами.

Diffraction selection of modes in planar backward-wave oscillators

We study the possibility of obtaining spatially coherent radiation in a planar backward-wave oscillator fed with a wide sheet electron beam by using natural diffraction divergence of the radiation. This method is efficient for Fresnel parameters being approximately equal to unity. To solve the two-dimensional boundary problem, the spatio-temporal approach is used, which allows one to determine both the self-excitation conditions and the parameters of the stationary generation regime and the boundaries of its stability. The developed theory is used to evaluate the possibility of achieving diffraction synchronization of the radiation in a high-power planar backward-wave oscillator operated in the 8-mm wavelength band and based on a sheet relativistic electron beam produced by the U-3 accelerator of the Institute of Nuclear Physics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia.

A model of coupled resonators for calculating the electromagnetic wave diffraction on planar one-dimensional Bragg gratings

The problem of excitation of a one-dimensional planar Bragg grating is studied. A model of coupled resonators is proposed for investigating the diffraction of electromagnetic waves on a waveguide surface with arbitrary corrugation profile. The model is based on the strict relationships derived from a solution to a two-dimensional boundary problem for the Helmholtz equation. A particular form of equations is presented for the case of a rectangular corrugation of plates in the grating.

Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions

We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green function under deformations of the domain reveals an integrable structure. The independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain. The solution to the Dirichlet boundary problem is expressed through the tau-function of the dispersionless Toda hierarchy. We also discuss a degenerate case of the Dirichlet problem on the plane with a gap. In this case the tau-function is identical to the partition function of the planar large $N$ limit of the Hermitean one-matrix model.

On the numerical solution of a free boundary identification problem

In this paper, we discuss an inverse problem of determining a part of the boundary of the depletion region of semiconductor devices. The existence of the unique solution is proved. The unknown boundaries as well as unknown potentials are parameterized by finite elements. Decoupling the discretized Laplace equation from the boundary conditions, we derive two algorithms to solves this problems. The obtained nonlinear systems of algebraic equations are solved by quasi-Newton methods. Numerical experiments, including comparisons of the behaviour of the proposed algorithms when applied to two-dimensional inverse boundary problems, are reported.

Investigation of heat and mass transfer in the evaporation zone of a heat pipe operating by the ‘inverted meniscus’ principle

A theoretical investigation of heat and mass transfer during evaporation in a wet capillary structure in contact with a heating wall is presented. The wall has grooves by which the vapour is removed. Setting and results of the numerical investigation of the mathematical model (stationary two-dimensional boundary problem with a free boundary corresponding to the surface of the interphase transition inside the capillary structure) are given. Calculations performed at different heat and hydraulic loads allow one to trace the phases of the process evolution: origin and growth of vapour zones (bubbles). The evaporation intensity distribution and functional characteristics of the capillary structure have been found.

Cubic B-spline collocation method for nonlinear static analysis of panels under mechanical and thermal loadings

This study presents the cubic B-spline collocation method for solution of one- and two-dimensional nonlinear boundary problems, describing the behavior of panels subjected to mechanical and thermal loadings. Snap-through of panels under pressure loading. and nonlinear thermally-induced response of panels are considered. Available results, yielded by other solution methods, were compared with those obtained by the proposed method. The accuracy, as well as efficiency of the present approach are verified.