Two-dimensional Boundary Value Problems Research Articles

Simulating the propagation of boundary-layer disturbances by solving boundary-value and initial-value problems

Abstract The article deals with the downstream propagation of small–amplitude disturbances of viscous incompressible laminar boundary layers, using the linearized equations for disturbance amplitudes. Two different methods are proposed. The first one solves a two-dimensional boundary-value problem, using a buffer-domain technique to mimic the outflow boundary condition. The second one solves a streamwise initial-value problem, using a spectral parabolization at each integration step. Both methods show good performance in simulating the propagation of the Tollmien–Schlichting waves and the Görtler vortices and can be applied to compute the spatial optimal disturbances.

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.

A two-dimensional boundary value problem of elliptic type with nonlocal conjugation conditions

Abstract We consider an elliptic boundary value problem with nonlocal conjugation conditions. An a priori estimate for its weak solution in an appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate, compatible with the smoothness of the input data, is obtained.

Compact Approximation of a Two-Dimensional Boundary Value Problem for Elliptic Equations of the Second Order with a Discontinuous Coefficient

Compact Approximation of a Two-Dimensional Boundary Value Problem for Elliptic Equations of the Second Order with a Discontinuous Coefficient

On Exact Solutions to Problems of Steady Flow of Incompressible Viscous Fluid in a Channel and in a Circular Pipe

Turning to the well-known Couette flow, we again, but now in a new formulation, consider the steady motion of an incompressible viscous fluid in a channel with two parallel flat walls, one of which is stationary while the other moves at a constant speed in its plane. In the classical formulation of this problem on a layered (laminar) fluid flow in a channel, the wall velocity and the constant pressure drop along the channel are independent quantities and are considered predetermined. However, when a solid body moves along the fluid surface, the fluid particles are set in motion. This occurs due to the arising pressure generated in the fluid caused by the movement of the solid body. Therefore, here the pressure or pressure drop is not predetermined but is determined as a result of solving the corresponding two-dimensional boundary value problem of a steady flow of a viscous fluid in a channel. For this, the corresponding boundary value problem of a fluid flow in the channel is formulated based on the linearized Navier-Stokes equations obtained from the general nonlinear Navier-Stokes equations for a steady flow of a viscous fluid neglecting the convective terms, which is true for small Reynolds numbers. Using the Fourier integral transform method, an exact (closed) solution to this boundary value problem is constructed; the velocity and pressure components are determined. Then it is shown that the resulting solution, different from the known Couette flow solution, also satisfies the original nonlinear Navier-Stokes equations, the continuity equation, and boundary conditions. A comparative analysis of the new solution with the Couette flow solution is carried out. The well-known Hagen-Poiseuille problem is also discussed from the point of view of establishing the conditions for the laminar axisymmetric and steady flow of an incompressible viscous fluid in a straight circular pipe According to the study's results, to implement laminar flow under the indicated conditions, the pressure drop along the pipe must be constant. The study demonstrates that this condition is simultaneously sufficient, and therefore, the necessary and sufficient condition is established for a laminar flow in a straight circular pipe.

A meshfree method for the solution of 2D and 3D second order elliptic boundary value problems in heterogeneous media

We propose a simple meshfree method for two-dimensional and three-dimensional second order elliptic boundary value problems in heterogeneous media based on equilibrated basis functions. The domain is discretized by a regular nodal grid, over which the degrees of freedom are defined. The boundary is also discretized by simply introducing some boundary points over it, independent from the domain nodes, granting the method the ability of application for arbitrarily shaped domains without the drawback of irregularity in the nodal grid. In heterogeneous media, the governing Partial Differential Equation (PDE) has non-constant coefficients for the partial derivatives, preventing Trefftz-based techniques to be applicable. The proposed method satisfies the PDE independent from the boundary conditions, as in Trefftz approaches. Meanwhile, by applying the weak form of the PDE instead of the strong form through weighted residual integration, the inconvenience of variable coefficients shall be resolved, since the bases will not need to analytically satisfy the PDE. The weighting in the weak form integrals is such that the boundary integrals vanish. The boundary conditions are thus collocated over the defined boundary points. Domain integrals break into algebraic combination of one-dimensional pre-evaluated integrals, thus omitting the numerical integration from the solution process. Each node corresponds to a local sub-domain called cloud. The overlap between adjacent clouds ensures integrity of the solution and its derivatives throughout the domain, an advantage with respect to C0 formulations.

A numerical two-scale approach for nonlinear hyperelastic beams and beam networks

We introduce a numerical framework for modeling hyperelastic slender trusses, oftentimes used as elementary building blocks in architected materials, which accounts for both the geometric nonlinearity inherent in thin structures and the nonlinear constitutive behavior of the base material. Akin to the FE2 method in homogenization, our approach is based on a formal, two-scale expansion. We decompose the three-dimensional (3D) description of a slender structure into a macroscale problem solving for the deformation of the beam center-line (based on an effective strain energy density, which depends on the stretching, bending, and torsional strains of the beam’s center-line) and a series of two-dimensional (2D) microscale boundary value problems (defined over the beam cross-section). We solve a series of 2D problems (covering a range of macroscopic strain combinations) for a given cross-sectional geometry and material distribution in an offline, pre-processing step. Using this pre-computed energy landscape, we solve the macroscopic boundary value problem within a geometrically exact, nonlinear discrete beam framework. We demonstrate the accuracy of this approach through a set of benchmark problems highlighting the nonlinear effects of the cross-sectional geometry and constitutive material, including material heterogeneity and pre-strains. We further illustrate how this technique is applied to truss-based architected materials consisting of networks of slender struts, which are typically made of polymeric base materials and thus allow for large nonlinear elastic deformation.

Two-dimensional Inverse Boundary Value Problem for a Third-order Pseudo-hyperbolic Equation with an Additional Integral Condition

In this paper we study an inverse boundary value problem with an unknown timedependent coefficient for a third-order pseudo-hyperbolic equation with an additional integral condition. The definition of the classical solution of the problem is given. The essence of the problem is that it is required together with the solution to determine the unknown coefficient. The problem is considered in a rectangular area. When solving the original inverse boundary value problem, the transition from the original inverse problem to some auxiliary inverse problem is carried out. The existence and uniqueness of a solution to an auxiliary problem are proved with the help of contracted mappings. Then the transition to the original inverse problem is again made, as a result, a conclusion is made about the solvability of the original inverse problem.

On eigenelements of a two-dimensional Steklov-type boundary value problem for the Lamé operator

In this paper, we study a two-dimensional Steklov-type boundary value problem for the Lamé operator in a half-strip, which is the limiting problem for a singularly perturbed boundary-value problem in a half-strip with a small hole. A theorem on the existence of eigenelements of the boundary value problem under study is proved. In particular, we obtain estimates for the eigenvalues expressed in terms of the Lamé constants and a parameter that determines the width of the half-strip, and refine the structure of the corresponding eigenfunctions, which determines their behavior as their argument move away from the base of the half-strip. Moreover, explicit expressions for the eigenvalues of the limiting boundary value problem are found up to the solution of a system of algebraic equations. The results obtained in this paper will make it possible to construct and rigorously justify an asymptotic expansion of the eigenvalue of a singularly perturbed boundary value problem in a half-strip with a small round hole in powers of a small parameter that determines the diameter of the hole.

Higher order approximation in exponential form based on half-step grid-points for 2D quasilinear elliptic BVPs on a variant domain

This paper reports a new fourth order Finite Difference Method (FDM) in exponential form for two-dimensional quasilinear boundary value problem of elliptic type (BVPE) with variant solution domain. Further, this discretization is extended to solve the system of quasilinear BVPEs. Following are the main highlights of the proposed FDM:• An unequal mesh 9-point compact stencil is used to approximate the solution. Half-step points are used to evaluate the known variables of this problem. The convergence theory is studied for unequal mesh to validate the fourth order convergence of the suggested FDM.• It is applicable to BVPE irrespective of coordinate systems. Various benchmark problems, for example, Poisson equation in cylindrical coordinates, Burgers’ equation, Navier-Stokes (NS) equations in cylindrical and rectangular coordinates, are solved to depict their fourth order convergence.• Numerical results confirm the accuracy, trustworthiness and acceptability of the suggested numerical algorithm. These results endorse the superiority of the proposed FDM over the previously existing techniques of Mohanty and Kumar (2014), Mohanty and Setia (2014), Priyadarshini and Mohanty (2021).

Local Refinement of Solutions in the Finite Element Method

We propose an approach to a posteriori improvement of approximations in projection methods for solving linear equations with selfadjoint positive definite operators. In particular, we consider the finite element method for two-dimensional boundary value problems.

A linear two-dimensional mathematical model for thin two-layer plates with partial shear interaction, with a view towards application to laminated glass

This work presents a consistent derivation, from three-dimensional linear elasticity, of a two-dimensional mathematical model describing the bending and in-plane stretching behaviours, under a general system of quasi-static distributed loads and prescribed support displacements, of thin two-layer plates with partial shear interaction. This layerwise model is specifically tailored for the requirements posed by the analysis of laminated glass plates commonly used in building structures (consisting of two thin glass layers bonded together by an adhesive interlayer). Our approach, based on Podio-Guidugli’s method of internal constraints, avoids mutually contradictory assumptions (not uncommon in the literature on structural mechanics) and yields a complete two-dimensional characterisation of displacement, strain and stress fields that exactly satisfy the field equations of three-dimensional linear elasticity and the boundary conditions at the end faces. The choice of generalised variables is designed to bring to light the following fundamental conclusion and physical insight: the resulting two-dimensional boundary value problem is a combination of the equations of Kirchhoff and Mindlin plates (with specified rigidities).Two examples illustrate the application of the proposed model: (i) the cylindrical bending of plate strips and (ii) a family of problems with Navier-type analytical solution. The solutions exhibit continuity across the whole range of zero, partial and full interaction between the layers. Moreover, the Navier-type solutions are consistently in close agreement with the results of three-dimensional finite element analyses. On the contrary, analogous results previously reported in the literature exhibit considerable deviations. An explanation for these discrepancies is discussed in detail.

H-Adaptive radial basis function finite difference method for linear elasticity problems

In this research work, the radial basis function finite difference method (RBF-FD) is further developed to solve one- and two-dimensional boundary value problems in linear elasticity. The related differentiation weights are generated by using the extended version of the RBF utilizing a polynomial basis. The type of the RBF is restricted to polyharmonic splines (PHS), i.e., a combination of the odd m-order PHS phi (r)=r^m with additional polynomials up to degree p will serve as the basis. Furthermore, a new residual-based adaptive point-cloud refinement algorithm will be presented and its numerical performance will be demonstrated. The computational efficiency of the PHS RBF-FD approach is tested by means of the relative errors measured in ell _2-norm on several representative benchmark problems with smooth and non-smooth solutions, using h-adaptive, uniform, and quasi-uniform point-cloud refinement.

Source term model for elasticity system with nonlinear dissipative term in a thin domain

Abstract This article establishes an asymptotic behavior for the elasticity systems with nonlinear source and dissipative terms in a three-dimensional thin domain, which generalizes some previous works. We consider the limit when the thickness tends to zero, and we prove that the limit solution u ∗ {u}^{\ast } is a solution of a two-dimensional boundary value problem with lower Tresca’s free-boundary conditions. Moreover, we obtain the weak Reynolds-type equation.

Some Two-Dimensional Boundary-Value Problems of Filtration Dynamics for Models with Proportional Caputo Derivative

Some Two-Dimensional Boundary-Value Problems of Filtration Dynamics for Models with Proportional Caputo Derivative

A two-dimensional boundary value problem for one overdetermined system

In the half-plane, we consider a stationary system of the two-velocity hydrodynamics with one pressure and homogeneous divergent and inhomogeneous boundary conditions for two velocities. This system is overdetermined. The solution of this system is reduced to the sequential solution of two boundary value problems: the Stokes problem for one velocity and pressure and the overdetermined system for the other velocity. With an appropriate choice of function spaces, the existence and uniqueness of a generalized solution with an appropriate stability estimate has been proved.

Solving of multi-connected curvilinear boundary value problems by the fast PIES

This paper presents the fast parametric integral equations system (PIES) in solving two-dimensional (2D) multi-connected curvilinear potential boundary value problems. The fast PIES is a combination of the fast multipole technique with a modified binary tree and original PIES. It was successfully employed for exploring the efficiency of the method applied in modelling and solving single-connected 2D potential problems. The fast PIES allows reducing the time of numerical computations (CPU time), as well as RAM utilization. However, application to the multi-connected problems requires appropriate changes in the fast multipole computations due to the small distance between some segments in internal parts of the boundary. The method is demonstrated with the examples of curvilinear potential problems dealing with perforated plates (plates with many circular holes).

On a Two-Dimensional Boundary-Value Stefan-Type Problem Arising in Cryosurgery

In this paper, we present the formulation and a method for solving the problem of freezing living biological tissue with a flat circular cryoapplicator. The model is a two-dimensional boundary-value problem of Stefan type with nonlinear heat sources of a special type that provide the actually observed spatial localization of the temperature field. Some results of computer simulation are presented.

Poroelastodynamic responses of a dual-porosity dual-permeability material under harmonic loading

We solve analytically and numerically a two-dimensional boundary value problem for the poroelastodynamic responses of a dual-porosity dual-permeability material subjected to harmonic excitation. To derive the analytical solutions, we first apply Helmholtz decomposition to decompose displacement vectors into divergence- and curl-free components. Then we use the diagonalization method to solve a series of coupled partial differential equations. The numerical solution is obtained using the finite element method. We present an example showing that the analytical and numerical solutions are close to each other.