Two-dimensional Initial-boundary Value Problems Research Articles

Analysis of Varying Temperature Regimes in a Conductive Strip during Induction Heating under a Quasi-Steady Electromagnetic Field

Transition processes in a steel conductive strip are analyzed during its induction heating under a quasi-steady electromagnetic field. In particular, the temperature field in the strip is studied. A method of solving corresponding initial boundary problems in a two-dimensional mathematical model for differential equations of electrodynamics and heat conduction is developed. The Joule heat and the temperature are determined with a high level of accuracy. The defining functions are the temperature and component of the magnetic field intensity vector tangent to the bases and end planes of the strip. To find them, we use cubic approximation of the defining functions’ distribution along the thickness coordinate. The original two-dimensional initial boundary value problems for the defining functions are reduced to one-dimensional initial boundary value problems on their integral characteristics. General solutions for these problems are obtained using the finite integral transformation by the transverse variable and the Laplace transform of the integral by time. Integral characteristics’ expressions are represented as convolutions for functions that describe homogeneous solutions of one-dimensional initial boundary value problems and limiting values of defining functions on the bases and end planes of the strip. The change of temperature under a varying regime in the dimensionless Fourier time and temperature distribution over the strip cross-section in a steady state depending on the parameters of induction heating and the Biot number are numerically analyzed. Varying and constant temperature regimes of the strip under conditions of the near-surface and continuous induction heating are studied.

Визначення двовимірних нестаціонарних температурних полів у плитах та панелях з плоскопаралельними межами за наявності джерел тепла

Non-stationary two-dimensional problems of thermal conductivity for plates and panels with plane-parallel boundaries in the presence of volume-distributed non-stationary heat sources are formulated. A method of constructing a solution to the formulated heat conduction problems for the considered bodies is proposed. The technique uses the approximation of the temperature distribution in both elements by the thickness variable by a cubic polynomial. The coefficients of the approximation polynomial are given through the integral over the thickness variable temperature characteristics and the conditions for the boundary values of the temperature on the outer surfaces of the plate and panel. As a result, the initial two-dimensional initial boundary value problems for the temperature for the plate and panel are reduced to one-dimensional initial boundary value problems for the integral temperature characteristics. To construct the solution of the initial-boundary value problem for the integral characteristics of the temperature in the case of a plate infinite in longitudinal and transverse coordinates, the integral Laplace transform in time and the integral Fourier transform in the longitudinal coordinate were used. The solution of the problem on the integral temperature characteristics in the case of the panel is found using the integral Laplace transform in time and the finite integral transform in the transverse coordinate. Expressions of integral temperature characteristics for the plate and panel are obtained in the form of convolutions of functions corresponding to homogeneous solutions of initial-boundary value problems for integral temperature characteristics and functions describing the available non-stationary heat sources in these bodies and given surface temperature values. The general solutions of the two-dimensional initial boundary value problems of thermal conductivity for slabs and panels are recorded for the presence of arbitrarily variable spatial coordinates of non-stationary heat sources and the conditions of convective heat exchange with the external environment on the surfaces of the considered bodies.

MODELING OF THERMO-ELASTIC-VISCO-PLASTIC DEFORMATION OF SHALLOW METAL-COMPOSITE SHELLS USING THE REFINED THEORY OF BENDING

The dynamic problem of thermo-elastic-viscous-plastic deformation of shallow composite shells is formulated using the refined theory of bending. In this case, tangential displacements along the thickness of constructions are approximated by polynomials of the third and higher orders, and the deflection does not depend on the transverse coordinate. Normal transverse stresses in the composition have a linear distribution over the thickness. The temperature over the thickness of the curved panels is approximated by a 7th order polynomial. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The constructions are multidirectionally reinforced with continuous fibres. The composition materials are isotropic; their plastic deformation is described by the flow theory with isotropic hardening, and the loading function depends on temperature and strain rate. The numerical solution of the coupled nonlinear two-dimensional initial-boundary value problem is obtained using an explicit scheme. Elastic-viscous-plastic, thermo-elastic-plastic and thermo-elastic-viscous-plastic response of a metal-composite cylindrical shallow shell of a rectangular shape with an orthogonal 2D-reinforcement structure has been studied. The construction is frontally loaded with an air blast wave. It has been demonstrated that curved metal-composite panels under such loading must be calculated, taking into account the temperature response that occurs in them and the sensitivity of the plastic properties of the composite materials to the rate of their deformation. In this case, it is necessary to apply the refined theory of bending of shallow shells, and not its simplest version, the Ambartsumian theory. A more intense thermomechanical response of a curved panel is observed when it is dynamically loaded from the convex front surface. In this case, at individual points of the construction, the temperature can briefly increase by 200 °C. It has been found that under dynamic loading of shallow shells from the side of any face-surface, they snapping towards the concavity of thin-walled structures.

Numerical analysis of heating by a current pulse of a niobium nitride membrane in its longitudinal section

A numerical calculation of the evolution of the temperature distribution in the longitudinal section of a niobium nitride membrane when it is heated by an electric current pulse is performed. Mathematical modeling was carried out on the basis of a two-dimensional initial-boundary value problem for an inhomogeneous heat equation. In the initial boundary value problem, it was taken into account that current and potential contacts to the membrane serve simultaneously as contacts for heat removal. The case was considered for the third from the left and the first from the right initial-boundary value problem. Analysis of the numerical solution showed that effective heat removal from the membrane can be provided by current-carrying and potential clamping contacts made, for example, of beryllium bronze. This makes it possible to study the current-voltage characteristics of superconducting membranes near the critical temperature of the transition to the superconducting state by currents close to the critical density without significant heating.

Boundedness in a haptotactic cross-diffusion system modeling oncolytic virotherapy

This paper is concerned with a haptotactic cross-diffusion system as the model for oncolytic virotherapy, which describes the influence of the extracellular matrix (ECM) taxis over the tumor-oncolytic virus interaction in the form of haptotaxis of both cancer cells and oncolytic virus toward higher ECM densities. The main results assert the global boundedness of solutions to an associated spatially two-dimensional initial-boundary value problems under suitably assumptions on the system parameters.

Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)

We propose and study a class of parabolic-ordinary differential equation models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumour microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature constituting a core difference to both classical Keller–Segel chemotaxis systems and Chaplain–Lolas type chemotaxis–haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.

Simplified Mathematical Model for the Description of Anomalous Migration of Soluble Substances in Vertical Filtration Flow

Since the use of the fractional-differential mathematical model of anomalous geomigration process based on the MIM (mobile–immoble media) approach in engineering practice significantly complicates simulations, a corresponding simplified mathematical model is constructed. For this model, we state a two-dimensional initial-boundary value problem of convective diffusion of soluble substances under the conditions of vertical steady-state filtration of groundwater with free surface from a reservoir to a coastal drain. To simplify the domain of simulation, we use the technique of transition into the domain of complex flow potential through a conformal mapping. In the case of averaging filtration velocity over the domain of complex flow potential, an analytical solution of the considered problem is obtained. In the general case of a variable filtration velocity, an algorithm has been developed to obtain numerical solutions. The results of process simulation using the presented algorithm shows that the constructed mathematical model can be efficiently used to simplify and accelerate modeling process.

Методика побудови розв’язку двовимірної задачі електродинаміки для електропровідного тіла з плоскопаралельними межами за дії зов-нішнього нестаціонарного електромагнітного поля

A two-dimensional initial-boundary value problem of electrodynamics for an electroconductive non-ferromagnetic body with planar-parallel boundaries is formulated under the action of an external non-stationary electromagnetic field. The electromagnetic field is given by the values of the inhomogeneous longitudinal coordinate of the component of the magnetic field intensity vector at its bases. To construct a general solution of the formulated initial boundary value problem under such electromagnetic action, a cubic approximation by thickness coordinate and an integral Fourier transform along a longitudinal coordinate was used for a key function — a given component of the magnetic field intensity vector. As a result, the initial two-dimensional initial boundary value problem for the key function is reduced to the one-dimensional initial boundary value problem. This initial boundary value problem is on the integral key function. These characteristics are the functions of time variable and the thickness coordinate. The coefficients of the polynomial approximating the key function are given by the transformants of the integral characteristics of the key function and given its values on the bases of the body as corresponding functions of time and longitudinal coordinate. The general solutions of the one-dimensional initial boundary value problem are obtained as a convolution of functions. These functions describe homogeneous solutions and key function boundary values on the bases of the body. Using the inverse Fourier transforms, the solution of the original electrodynamics problem under the action of an arbitrary variable in time and by the longitudinal coordinate of a non-stationary electromagnetic field is written. On the basis of such a common solution, as a partial case, we also write the solutions of the original two-dimensional initial boundary value problem under the action of a stationary harmonic in time variable electromagnetic field. In order to improve the accuracy of the approximate solution of the problem under consideration, an independent approximation of the corresponding component of the electric field intensity vector along the thickness coordinate is proposed. Systems of equations are formulated to determine the integral characteristics of this component, both under the action of an arbitrary variable in time and for a longitudinal coordinate non-stationary and the stationary electromagnetic field.

An evolution problem related to nonlinear 2d-magnetoelasticity

There is studied a two-dimensional initial–boundary-value problem for the Lamé and Maxwell equations coupled through the nonlinear magnetoelastic effect. We prove the existence and uniqueness result.

Navier wall law for nonstationary viscous incompressible flows

We study the Navier wall law for the two-dimensional initial boundary value problem of the Navier–Stokes equations in a domain with a rough boundary. The Navier wall law is verified for the initial data in C1 class under the natural compatibility condition. Our proof relies on the boundary layer analysis and the L∞ theory of the Navier–Stokes equations in the half space.

Finite difference approximation for parabolic interface problem with time-dependent coefficients

The convergence of difference scheme for two-dimensional initial boundary value problem for the heat equation with concentrated capacity and time-dependent coefficients of the space derivatives, is considered. An estimate of the rate of convergence in a special discrete W~12,1/2 Sobolev norm, compatible with the smoothness of the coefficients and solution, is proved.

A coupled ductile fracture phase-field model for crystal plasticity

Nowadays crack initiation and evolution play a key role in the design of mechanical components. In the past few decades, several numerical approaches have been developed with the objective to predict these phenomena. The objective of this work is to present a simplified, nonetheless representative phenomenological model to predict the crack evolution of ductile fracture in single crystals. The proposed numerical approach is carried out by merging a conventional elasto-plastic crystal plasticity model and a phase-field model modified to predict ductile fracture. A two-dimensional initial boundary value problem of ductile fracture is introduced considering a single-crystal setup and Nickel-base superalloy material properties. The model is implemented into the finite element context subjected to a quasi-static uniaxial tension test. The results are then qualitatively analyzed and briefly compared to current benchmark results in the literature.

On the Variational Formulation of the Extended Thick Anisotropic Shells Theory of I. N. Vekua Type

A new variant of the I. N. Vekua – A. A. Amosov extended theory of thick anisotropic shells is constructed on the groundwork of the Lagrange variational formalism of analytical mechanics of continua and the dimensional reduction approach. The shell model consists in the set of field variables, the surface Lagrangiandensity, and the constraint equations defined on the two-dimensional manifold corresponding to the base surface. The supplementary constraints are derived from the boundary conditions on shell's faces. The field variables of the first kind are biorthogonal expansion's coefficients for the displacement vector. This constrained variational problem is solving by the Lagrange multipliers method that results the two-dimensional initial-boundary value problem of the general shell theory.

Numerical approximation of a two-dimensional parabolic time-dependent problem containing a delta function

The convergence of a difference scheme for a two-dimensional initial-boundary value problem for the heat equation with concentrated capacity and time-dependent coefficients of the space derivatives is considered. An estimate of the rate of convergence in a special discrete W˜22,1 Sobolev norm, compatible with the smoothness of the coefficients and the solution, is proved.

Determining the internal boundary of a region in the two-dimensional initial–boundary-value problem for the heat equation

The article investigates the reconstruction of the internal boundary of a two-dimensional region in the two-dimensional initial---boundary-value problem for the homogeneous heat equation. The initial values for the determination of the internal boundary are provided by a boundary condition of second kind on the external boundary and the solution of the initial---boundary-value problem at finitely many points inside the region. The inverse problem is reduced to solving a system of integral equations nonlinear in the function describing the sought boundary. An iterative numerical procedure is proposed involving linearization of integral equations.

Numerical solution of two-dimensional reaction–diffusion Brusselator system

In this paper, polynomial based differential quadrature method (DQM) is applied for the numerical solution of a class of two-dimensional initial-boundary value problems governed by a non-linear system of partial differential equations. The system is known as the reaction–diffusion Brusselator system. The system arises in the modeling of certain chemical reaction–diffusion processes. In Brusselator system the reaction terms arise from the mathematical modeling of chemical systems such as in enzymatic reactions, and in plasma and laser physics in multiple coupling between modes. The numerical results reported for three specific problems. Convergence and stability of the method is also examined numerically.

Time-dependent magnetohydrodynamic flow induced by non-coaxial rotations of a non-torsionally oscillating porous plate and a third-order fluid at infinity

Time-dependent flow, induced by the rotation of a fluid in an infinitely extending space and a suddenly inserted non-coaxially rotating infinite plate, is investigated. The plate is porous, allows suction or blowing through it, and simultaneously oscillates non-torsionally in its own plane. The fluid is considered as an electrically conducting third-order non-Newtonian fluid. A magnetic field perpendicular to the plate is exerted to the fluid. To solve the emerging high-order two-dimensional initial-boundary value problem, a coordinate transformation is employed to transform the infinite physical space to a finite computational domain and a suitable finite difference method is used for numerical simulation. The time series of the flow velocity, the structure of the boundary layer near the plate, the influences of the exerted magnetic field, the suction/blowing velocity through the plate and a slip boundary condition at the plate on the flow are graphically presented and discussed. The nature of the shear stress engendered due to the flow is also investigated. In particular, numerical results for the third-order fluid are compared with those of a corresponding Newtonian fluid.

The two-dimensional reaction–diffusion Brusselator system: a dual-reciprocity boundary element solution

The dual-reciprocity boundary element method is applied for the numerical solution of a class of two-dimensional initial-boundary value problems governed by a non-linear system of partial differential equations. The system, known as the reaction–diffusion Brusselator, arises in the modeling of certain chemical reaction–diffusion processes. Numerical results are presented for some specific problems.

Convergence study of the Chorin-Marsden formula

Using the fundamental solution of the heat equation, we give an expression of the solutions to two-dimensional initial-boundary value problems of the Navier-Stokes equations, where the vorticity is expressed in terms of a Poisson integral, a Newtonian potential, and a single layer potential. The density of the single layer potential is the solution to an integral equation of Volterra type along the boundary. We prove there is a unique solution to the integral equation. One fractional time step approximation is given, based on this expression. Error estimates are obtained for linear and nonlinear problems. The order of convergence is 1/4 for the Navier-Stokes equations. The result is in the direction of justifying the Chorin-Marsden formula for vortex methods. It is shown that the density of the vortex sheet is twice the tangential velocity for the half plane, while in general the density differs from it by one additional term.

RATE OF DRYING AND STRESSES IN THE FIRST PERIOD OF DRYING*

ABSTRACT The paper presents a computer simulated processes and illustrate how the drying induced stresses are influenced by the rate of drying. It is shown that the moisture transport coefficient, and thus the rate of drying, depends on the thermal state of the drying material, defined by the wet-bulb temperature. Through these simulated processes one can observe the evolution of the moisture content and stress distributions during drying at constant, but in each process different, wet-bulb temperatures. A convective drying process of a bar with rectangular cross-section is considered as example, and a two-dimensional initial-boundary value problem is solved numerically with the use of the finite element method. The numerical results are visualised in spatial diagrams.