Two-dimensional Boundary Value Problems Research Articles

Система компьютерного моделирования для решения двумерных краевых задач с использованием бессеточного похода

Система компьютерного моделирования для решения двумерных краевых задач с использованием бессеточного похода

Solution of the Problem of Free Vibrations of a Nonthin Orthotropic Shallow Shell of Variable Thickness in the Refined Statement

We consider the problem of investigation of the spectrum of natural vibrations of a nonthin orthotropic shallow shell variable in two coordinate directions of thickness in the nonclassical statement. The approach to the solution of the obtained two-dimensional boundary-value problem is based on its reduction (by the method of spline-approximation of the unknown functions along one coordinate direction) to the one-dimensional problem with its subsequent solution. We study different cases of boundary conditions imposed on the contours of the shell. We also perform the comparison and analysis of the natural frequencies and modes of vibrations of orthotropic shells of constant and variable thickness.

Analysis and explicit solvability of degenerate tensorial problems

We study a two-dimensional boundary value problem described by a tensorial equation in a bounded domain. Once its more general definition is given, we conclude that its analysis is linked to the resolution of an overdetermined hyperbolic problem and hence present some discussions and considerations. Secondly, for a simplified version of the original formulation, which leads to a degenerate problem on a rectangle, we prove the existence and uniqueness of a solution under proper assumptions on the data.

Convergence Results and Optimal Control for a Class of Hemivariational Inequalities

The present paper deals with new results in the study of a class of elliptic hemivariational inequalities in reflexive Banach spaces. We start with an existence and uniqueness result. We complete it with a convergence result which describes the dependence of the solution with respect to the data. To this end, we use the notion of Mosco convergence for the set of constraints. Next, we formulate an optimal control problem for which we prove the existence of optimal pairs and state a convergence result. Finally, we exemplify the use of our results in the study of a two-dimensional boundary value problem which describes the frictionless contact of an elastic body with two reactive foundations.

Improved Kansa RBF method for the solution of nonlinear boundary value problems

We apply the Kansa–radial basis function (RBF) collocation method to two-dimensional nonlinear boundary value problems. In it, the solution is approximated by a linear combination of RBFs and the governing equation and boundary conditions are satisfied in a collocation sense at interior and boundary points, respectively. The nonlinear system of equations resulting from the Kansa–RBF discretization for the unknown coefficients in the RBF approximation is solved by directly applying a standard nonlinear solver. In a natural way, the value of the shape parameter in the RBFs employed in the approximation may be included in the unknowns to be determined. The numerical results of several examples are presented and analyzed.

A complex variable boundary element method for solving a steady-state advection–diffusion–reaction equation

An accurate complex variable boundary element method is proposed for the numerical solution of two-dimensional boundary value problems governed by a steady-state advection–diffusion–reaction equation. With the aid of the Cauchy integral formulae, the task of constructing a complex function which gives the solution of the boundary value problem under consideration is reduced to solving a system of linear algebraic equations. The method is applied to solve several specific problems which have exact solutions in closed form and a problem of practical interest in engineering.

Analysis and control of a nonlinear boundary value problem

We consider a nonlinear two-dimensional boundary value problem which models the frictional contact of a bar with a rigid obstacle. The weak formulation of the problem is in the form of an elliptic variational inequality of the second kind. We establish the existence of a unique weak solution to the problem, then we introduce a regularized version of the variational inequality for which we prove existence, uniqueness and convergence results. We proceed with an optimal control problem for which we prove the existence of an optimal pair. Finally, we consider the corresponding optimal control problem associated to the regularized variational inequality and prove a convergence result.

Numerical study of a two-dimensional mathematical model with variable heat exchange coefficient which arises in cryosurgery

We formulate and numerically solve a two-dimensional boundary value problem of Stefan type with nonlinear heat sources of a special kind and a variable heat exchange coefficient. The model under study arises in cryosurgery in the process of freezing some living biological tissue by a cryoinstrument of cylindrical shape placed on the surface of the tissue. The model takes into account the actually observed effect of spatial localization of heat. Some results of the computer simulation are presented.

A critical assessment of flux and source term closures in shallow water models with porosity for urban flood simulations

The validity of flux and source term formulae used in shallow water models with porosity for urban flood simulations is assessed by solving the two-dimensional shallow water equations over computational domains representing periodic building layouts. The models under assessment are the Single Porosity (SP), the Integral Porosity (IP) and the Dual Integral Porosity (DIP) models. 9 different geometries are considered. 18 two-dimensional initial value problems and 6 two-dimensional boundary value problems are defined. This results in a set of 96 fine grid simulations. Analysing the simulation results leads to the following conclusions: (i) the DIP flux and source term models outperform those of the SP and IP models when the Riemann problem is aligned with the main street directions, (ii) all models give erroneous flux closures when is the Riemann problem is not aligned with one of the main street directions or when the main street directions are not orthogonal, (iii) the solution of the Riemann problem is self-similar in space-time when the street directions are orthogonal and the Riemann problem is aligned with one of them, (iv) a momentum balance confirms the existence of the transient momentum dissipation model presented in the DIP model, (v) none of the source term models presented so far in the literature allows all flow configurations to be accounted for(vi) future laboratory experiments aiming at the validation of flux and source term closures should focus on the high-resolution, two-dimensional monitoring of both water depth and flow velocity fields.

Error analysis of a compact finite difference method for fourth-order nonlinear elliptic boundary value problems

This paper is concerned with a compact finite difference method with non-isotropic mesh sizes for a two-dimensional fourth-order nonlinear elliptic boundary value problem. By the discrete energy analysis, the optimal error estimates in the discrete L2, H1 and L∞ norms are obtained without any constraint on the mesh sizes. The error estimates show that the compact finite difference method converges with the convergence rate of fourth-order. Based on a high-order approximation of the solution, a Richardson extrapolation algorithm is developed to make the final computed solution sixth-order accurate. Numerical results demonstrate the high-order accuracy of the compact finite difference method and its extrapolation algorithm in the discrete L2, H1 and L∞ norms.

Solution of Stress-Strain Problems for Complex-Shaped Plates in a Refined Formulation

A numerical-analytical approach to solving problems on the stress–strain state of quadrangular plates of complex shape is proposed. The governing system of equations is presented in new orthogonal coordinates using transformations that take into account the plate geometry. A two-dimensional boundary-value problem, which is described by a system of partial differential equations derived with the spline-collocation method, is reduced to a one-dimensional one that is solved by the stable numerical discrete-orthogonalization method. The numerical results obtained for plates in the form of a trapezium and parallelogram are compared with the data obtained by other methods. The approach makes it possible to calculate deflections of quadrangular plates of complex shape made of anisotropic materials

Evaluation of effective material properties in magneto-electro-elastic composite materials

The meshless local integral equation method is developed to analyze general two-dimensional boundary value problems in size-dependent magnetoelectroelastic solids. A consistent theory is developed for size dependent magnetoelectroelasticity. The strain gradients are considered in the constitutive equations for electric displacement and magnetic induction. The governing equations are derived with the corresponding boundary conditions using the variational principle. The local integral equations are subsequently derived and the meshless moving least square (MLS) numerical method is implemented to solve these equations.

Solving Elliptic Equations by the Generalized Point-Sources Method

The aim of this work is to develop a new universal numerical method for solving boundary value problems for linear elliptic equations. This method is based on the transformation of the original mathematical physics equation to a simpler inhomogeneous equation with the known fundamental solution. From this equation the transition is carried out to an inhomogeneous integral equation with a kernel, expressed by the known fundamental solution. Obtained integral equation together with boundary conditions is solved numerically. The result is obtained approximate solution, the field potential, in an analytical form. This allows not only to find approximate value of the field potential at any point in the solutions domain, but also to differentiate this potential and without significant loss of accuracy. This property of the worked out numerical method distinguishes it from traditional numerical methods for solving boundary value problems, such as the finite element method. To confirm the effectiveness of the proposed numerical method the two-dimensional and three-dimensional boundary value problems with the pre known solutions were resolved. The dependences of the numerical solutions error on the number of linear equations in the resulting system were obtained. It is shown that even at a small number of equations in the system, a few hundred, the solutions accuracy being achieved within hundredths of a percent. The results obtained in this work show that a physical field described by any linear elliptic equation, virtually, can be represented as a superposition of point sources fields, satisfying a simpler equation, the solution of which is by the method of point sources of the field. Therefore the numerical method, presented in this paper, can be considered as the generalized point sources method, which allows to greatly extend the scope of the traditional the point sources method in solving applications for modeling fields of different physical nature in various types technical devices.

The MLPG in gradient theory for size-dependent magnetoelectroelasticity

The strain gradient magnetoelectroelasticity is applied to solve two-dimensional boundary value problems. The electric and magnetic field-strain gradient coupling is considered in constitutive equations. The meshless local Petrov-Galerkin (MLPG) is developed to solve general problems. All field quantities are approximated by the moving least-squares (MLS) scheme. Effective material properties for a piezomagnetic matrix with regularly distributed piezoelectric fibres of a circular cross section and coating layer are presented.

Tearing instability in a tokamak with a noncircular cross section

Using a straight-column model to describe tokamak plasma with a noncircular cross section, it is shown how to (i) find the boundary of tearing instability from the condition of existence of a magnetohydrodynamic plasma equilibrium different from that of a straight cylinder by solving a two-dimensional linear boundary-value problem with a second-order equation with respect to the flux coordinate and (ii) find the spatial structure of the tearing mode and the corresponding effective Δ' when there is only one resonance magnetic surface in the plasma for a given axial wavenumber by solving some kind of a boundary-value problem for the perturbation. The proposed approach is illustrated by numerical calculations for the case of an elliptical cross section as an example.

Surface and internal waves: The two-dimensional problem on forward motion of a body intersecting the interface between two fluids

A linear two-dimensional boundary value problem, that describes steady-state surface and internal waves due to the forward motion of a body in a fluid consisting of two superposed layers with different densities, is considered. The body is fully submerged and intersects the interface between the two layers. Two well-posed formulations of the problem are proposed in which, along with the Laplace equation, boundary conditions, coupling conditions on the interface, and conditions at infinity, a pair of supplementary conditions are imposed at the points where the body contour intersects the interface. In one of the well-posed formulations (where the differences between the horizontal momentum components are given at the intersection points), the existence of the unique solution is proved for all values of the parameters except for a certain (possibly empty) nowhere dense set of values.

Mixed meshless local Petrov–Galerkin collocation method for modeling of material discontinuity

A mixed meshless local Petrov---Galerkin (MLPG) collocation method is proposed for solving the two-dimensional boundary value problem of heterogeneous structures. The heterogeneous structures are defined by partitioning the total material domain into subdomains with different linear-elastic isotropic properties which define homogeneous materials. The discretization and approximation of unknown field variables is done for each homogeneous material independently, therein the interface of the homogeneous materials is discretized with overlapping nodes. For the approximation, the moving least square method with the imposed interpolation condition is utilized. The solution for the entire heterogeneous structure is obtained by enforcing displacement continuity and traction reciprocity conditions at the nodes representing the interface boundary. The accuracy and numerical efficiency of the proposed mixed MLPG collocation method is demonstrated by numerical examples. The obtained results are compared with a standard fully displacement (primal) meshless approach as well as with available analytical and numerical solutions. Excellent agreement of the solutions is exhibited and a more robust, superior and stable modeling of material discontinuity is achieved using the mixed method.

FEM formulation for size-dependent theory with application to micro coated piezoelectric and piezomagnetic fiber-composites

The finite element method (FEM) is developed to analyze general two-dimensional boundary value problems in size-dependent magnetoelectroelastic solids. The size-effect phenomena in micro/nano electronic structures are described by the strain gradient effect. The electric and magnetic field-strain gradient coupling is considered in the constitutive equations of the material and the governing equations are derived with the corresponding boundary conditions by virtue of the variational principle. The FEM formulation is subsequently developed and implemented for strain-gradient magnetoelectroelasticity and a couple of numerical examples are presented to illustrate the strain gradient effect on the fields.

Concept of modeling uncertainly defined shape of the boundary in two-dimensional boundary value problems and verification of its reliability

The paper presents the concept of modeling uncertainly defined shape of the boundary in two-dimensional boundary value problems described by Laplace equation. The authors propose generalization of parametric integral equations system (PIES) in modeling and solving the problems. Previously, PIES was successfully applied to model precisely defined problems. Pseudo-spectral method and modified interval arithmetic are used to define the boundary of the problem and to obtain numerical solution of generalized PIES. We examine direct application of interval Bézier curves in PIES and then we propose our new concept of modeling of the shape of the boundary to improve the reliability of solutions.

Analysis of the Stress–Strain State of Inhomogeneous Hollow Cylinders

The stress-strain state of an inhomogeneous hollow cylinder with different boundary conditions at the ends is analyzed using the three-dimensional theory of elasticity. Spline collocation is used to reduce the two-dimensional boundary-value problem to a boundary-value problem for a system of ordinary differential equations of high order with respect to the radial coordinate, which is solved with the stable discrete-orthogonalization method. The results obtained using the spline-collocation, Fourier-series, and finite-element methods are compared materials, and the consideration of three-dimensional effects in thick-walled structural members necessitate studying hollow cylindrical structures in three dimensions. The stress-strain analysis of thick-walled structures based on the three-dimensional theory of elasticity involves severe difficulties associated with the complexity of the starting systems of partial differential equations and the necessity to satisfy boundary conditions on the surface of an elastic body. These difficulties are even more severe when designing cylindrical elements made of anisotropic and inhomogeneous materials, such as functionally graded materials (FGM) with variable elastic characteristics. Modern technologies make it possible to produce structures with required smoothly varying elastic moduli. The physical and mechanical properties of FGMs based on various compositions are addressed in (3, 11-13). Of great practical interest and importance for fundamental research is the analysis of the stress-strain state of various structural members made of FGMs, including finite-length cylinders. In view of the above, it is necessary to determine the dynamic characteristics of such structural members in three dimensions. Due to great computational difficulties, there are only few publications on the