Numerical Solution Of Boundary Value Problems Research Articles

Formulation and Numerical Solution of Plane Problems of the Theory of Elasticity in Strains

This article is devoted to the formulation and numerical solution of boundary-value problems in the theory of elasticity with respect to deformations. Similar to the well-known Beltrami–Michell stress equations, the Saint-Venant compatibility conditions are written in the form of differential equations for strains. A new version of plane boundary-value problems in strains is formulated. It is shown that for the correctness of plane boundary value problems, in addition to the usual conditions, one more special boundary condition is required using the equilibrium equation. To discretize additional boundary conditions and differential equations, it is convenient to use the finite difference method. By resolving grid equations and additional boundary conditions with respect to the desired quantities at the diagonal nodal points, we obtained convergent iterative relations for the internal and boundary nodes. To solve grid equations, the elimination method was also used. By comparing with the Timoshenko–Goodyear solution on the tension of a rectangular plate with a parabolic load, the validity of the formulated boundary value problems in strains and the reliability of the numerical results are shown. The accuracy of the results has been increased by an average of 15%.

Learning Data for Neural-Network-Based Numerical Solution of PDEs: Application to Dirichlet-to-Neumann Problems

We propose neural-network-based algorithms for the numerical solution of boundary-value problems for the Laplace equation. Such a numerical solution is inherently mesh-free, and in the approximation process, stochastic algorithms are employed. The chief challenge in the solution framework is to generate appropriate learning data in the absence of the solution. Our main idea was to use fundamental solutions for this purpose and make a link with the so-called method of fundamental solutions. In this way, beyond the classical boundary-value problems, Dirichlet-to-Neumann operators can also be approximated. This problem was investigated in detail. Moreover, for this complex problem, low-rank approximations were constructed. Such efficient solution algorithms can serve as a basis for computational electrical impedance tomography.

Accuracy of a DtN Finite Element Approach for the Elastic Half-Space

In this work, the accuracy of a DtN finite element approach is numerically assessed. This procedure allows an efficient and accurate numerical solution of boundary-value problems of axisymmetric elasticity in semi-infinite domains. For this kind of problems, no explicit closed-form expression for the associated DtN map exists, so a suitable semi-analytical approximation of it, in series form, is used to impose exact boundary conditions on a semi-spherical artificial boundary. The procedure is tested by solving the classical Boussinesq problem, whose exact solution is known analytically. The computed numerical solution is compared to the analytical solution, achieving an excellent agreement between both solutions, both for displacements and stresses. The convergence of the analytical solution to the numerical solution is numerically demonstrated, in terms of artificial boundary location, series truncation order and finite element mesh size.

ЧИСЕЛЬНИЙ АНАЛІЗ РОЗВ’ЯЗКІВ ДВОВИМІРНИХ ЗАДАЧ ТЕПЛОПРОВІДНОСТІ ЗА БЕЗСІТКОВОЮ СХЕМОЮ З ВИКОРИСТАННЯМ ФУНДАМЕНТАЛЬНИХ І ЗАГАЛЬНИХ РОЗВ’ЯЗКІВ

У статті наводиться аналіз чисельних розв’язків двовимірних задач теплопровідності за безсітковою схемою, які отримані з використанням фундаментальних і загальних розв’язків модифікованого рівняння Гельмгольца. У роботі використовувався безсітковий метод, який ґрунтується на комбінації методу подвійного заміщення з використанням радіальних базисних функцій і методу частинних розв’язків. Порівняльний аналіз розв’язків крайових задач продемонстрований на прикладі двох тестових задач. Були отримані чисельні розв’язки двовимірних нестаціонарних задач теплопровідності з використанням фундаментального і загального розв’язків для різного числа інтерполяційних вузлів. Були визначені середньоквадратичні похибки розв’язків розглянутих задач, а також побудовані порівняльні графіки залежності середньоквадратичної похибки від числа інтерполяційних вузлів.

Strength of Elastoplastic Plates Containing Square Holes (Inclusions) and Cuts (Thin Inclusions) Under Uniaxial Tension

We study the influence of stress concentrators (square hole, cut, or perfectly rigid inclusion) in a plate on its strength under uniaxial tension with the help of the numerical solution of boundary-value problems of the theory of small elastoplastic deformations for a linearly hardened material with regard for its unloading.

Boundary Element Solution for Dynamics of Inhomogeneous Poroviscoelastic Beam

The present paper is dedicated to numerical solution of boundary-value problems for piecewise homogeneous solids in terms of linear three-dimensional poroviscoelasticity. Mathematical model of poroviscoelastic material is based on Biot's model of poroelasticity. Viscoelastic effects refer to a skeleton of porous material and are described through the correspondence principle. Standard linear solid model is employed. In order to study the boundary-value problem boundary integral equations method is applied, and to find their solutions boundary element method for obtaining numerical solutions. A direct version of the BIE method is developed. The boundary-element scheme is constructed using: regularized BIE’s, a matched element-by-element approximation, adaptive numerical integration in combination with a singularity-reducing algorithm. The solution of the original problem is constructed in Laplace transforms, with the subsequent application of the algorithm for numerical inversion. Modified Durbin’s algorithm of numerical inversion of Laplace transform is applied to perform solution in time domain. The problem of poroviscoelastic prismatic solid clamped from one end and free at another is considered. The solid is composed of two subdomains. Heaviside-type load is applied to a free end of the solid. Viscosity parameter influence on dynamic responses of displacements, pore pressure and tractions is studied. Numerical results for displacements and pore pressure when subdomains are modelled with different viscoelastic properties are presented.

A general result for the magnetoelastic response of isotropic suspensions of iron and ferrofluid particles in rubber, with applications to spherical and cylindrical specimens

This paper puts forth an approximate solution for the effective free-energy function describing the homogenized (or macroscopic) magnetoelastic response of magnetorheological elastomers comprised of non-Gaussian rubbers filled with isotropic suspensions of either iron or ferrofluid particles. The solution is general in that it applies to N=2 and 3 space dimensions and any arbitrary (non-percolative) isotropic suspension of filler particles. By construction, it is exact in the limit of small deformations and moderate magnetic fields. For finite deformations and finite magnetic fields, its accuracy is demonstrated by means of direct comparisons with full-field simulations for two prominent cases: (i) isotropic suspensions of circular particles and (ii) isotropic suspensions of spherical particles.With the combined objectives of demonstrating the possible benefits of using ferrofluid particles in lieu of the more conventional iron particles as fillers and gaining insight into recent experimental results, the proposed homogenization-based constitutive model is deployed to generate numerical solutions for boundary-value problems of both fundamental and practical significance: those consisting of magnetorheological elastomer specimens of spherical and cylindrical shape that are immersed in air and subjected to a remotely applied uniform magnetic field. It is found that magnetorheological elastomers filled with ferrofluid particles can exhibit magnetostrictive capabilities far superior to those of magnetorheological elastomers filled with iron particles. The results also reveal that the deformation and magnetic fields are highly heterogenous within the specimens and strongly dependent on the shape of these, specially for magnetorheological elastomers filled with iron particles. From an applications perspective, this evidence makes it plain that attempts at designing magnetrostrictive devices based on magnetorheological elastomers need to be approached, in general, as structural problems, and not simply as materials design problems.

Realization of Exact Three-Point Difference Schemes for Nonlinear Boundary-Value Problems on the Semiaxis

New algorithmic realization of exact three-point difference schemes via the three-point difference schemes of high order of accuracy is proposed for the numerical solution of boundary-value problems for systems of nonlinear ordinary differential equations on the semiaxis. We study the existence and uniqueness of the solution of three-point difference schemes and estimate the rate of convergence. The results of numerical experiments are also presented.

Exponential difference schemes for solving boundary-value problems for convection-diffusion type equations

The paper considers the numerical solution of boundary-value problems for multidimensional convection-diffusion type equations (CDEs). Such equations are useful for various physical processes in solids, liquids and gases. A new approach to the spatial approximation for such equations is proposed. This approach is based on an integral transformation of second-order one-dimensional differential operators. A linear version of CDE was chosen for simplicity of the analysis. In this setting, exponential difference schemes were constructed, algorithms for their implementation were developed, a brief analysis of the stability and convergence was made. This approach was numerically tested for a two-dimensional problem of motion of metallic particles in water flow subject to a constant magnetic field.

Finite-Layer Method: Exact Numerical and Analytical Calculations of the Energy Release Rate for Unidirectional Composite Specimens in Double-Cantilever Beam and End-Notched Flexure Tests

Based on the finite-layer method, a method for evaluating the stress-strain state and energy release rate for specimens with delaminations in double-cantilever beam and end-notched flexure tests is proposed. Exact numerical solutions of boundary-value problems for the “stiff” systems of differential equations describing deformations of test specimens are obtained. The distributions of forces, moments, displacements, and rotations in the specimens and the distributions of normal and tangential stresses on their midline are presented. New closed-form expressions for these functions and for compliance of the specimens are developed. Calculation results for the energy release rate obtained by a numerical differentiation and from analytical relations are presented. Two new techniques for estimating the energy release rate are proposed: (1) using the calculated values of peak stress and jumps of displacements at the tip of delamination; (2) by evaluation of indeterminacy at the tip of delamination with the use of stresses and derivatives of stresses and displacements. The effect of the transverse shear and Poisson ratio on the results is estimated. A comparison of the numerical and analytical solutions obtained with known results and the ASTM standard is presented.

Analytical and numerical method of finite bodies for calculation of cylindrical orthotropic shell with rectangular hole

To solve the boundary-value problem for cylindrical orthotropic shell with sizeable rectangular hole we suggest analytical and numerical method of finite bodies. For determination of the stress state of orthotropic thin-walled cylinder we use a systemof equations that exactly satisfies the equilibrium equations of orthotropic cylindrical shell. Representation of the solutions is divided into basic and self-equilibrium state. For some loads of a shell we build the basic stress state. We obtain a countable number of resolving functions that exactly satisfy the equations of a shell and describe the self-equilibrium stress state. We develop the algorithm of the analytical and numerical solutions of boundary-value problem based on approximation of the stress state of a shell by finite sum of resolving functions and propose a universal way of reduction of all conditions of the contact parts of the enclosure and the boundary conditions to minimize the generalized quadratic forms. We establish criteria under which the construction of approximate solutions coincides with the exact one.

A convergence analysis of the numerical solution of boundary-value problems by using two-dimensional Haar wavelets

A convergence analysis of the numerical solution of boundary-value problems by using two-dimensional Haar wavelets

Theory of the formation of the energy spectra of reflected charged particles

The theory of the formation of the energy spectra of electrons and light ions reflected form a solid surface is constructed. It is shown that methods based on the numerical solution of boundary-value problems for the transport equation are the most effective. The boundary-value problem is solved based on the invariant embedding method. Equations for the reflection coefficients Rk(x, n, n0), describing the flux of atomic particles reflected in the direction n and having experienced k events of elastic scattering are obtained. The obtained results are compared with experimental data, the results of simulation using the Monte Carlo method, and the calculations of other authors.

On a Steffensen-like method for solving nonlinear equations

We study a generalization of Steffensen's method in Banach spaces. Our main aim is to obtain similar convergence as Newton's method, but without evaluating the first derivative of the operator involved. As motivation, we analyse numerical solutions of boundary-value problems approximated by the multiple shooting method that uses the proposed iterative scheme. Sufficient conditions for the semilocal convergence analysis of the method, including error estimates and the $$R$$R-order of convergence, are provided. Finally, the theoretical results are applied to a nonlinear system of equations related with the approximation of a Hammerstein-type integral equation.

Анализ разностной схемы аналога волнового уравнения с оператором дробного дифференцирования

Analysis of the difference scheme of boundary-value problem for the wave equation analogue is made. Explicit and implicit difference schemes for numerical solution of the first boundary-value problem for the wave equation analogue with Caputo fractional differentiation operator are investigated, and the stability criteria for these difference schemes are proved by the harmonic Fourier method. Estimates for eigenvalues of the operator of transition from one time layer to another are obtained. Computational experiment on the analysis of the given difference scheme has been performed for the example. The graphs of the numerical solution of the boundary-value problem for the wave equation with the operator of fractional differentiation having different values of parameters of fractional differentiation $\alpha$ and $\beta$ have been built. Change of the period of fluctuations under transition to a fractional derivative is established. On an example it is shown that parameters $\alpha$ and $\beta$ become managing directors.

Расчет параметров нагружения полого шара в условиях больших упругоползущих деформаций

Institute for Problems in Mechanics of RAS, 101-1, Vernadskogo ave., 119526, Moscow, Russia, murashkin@ipmnet.ru,evmurashkin@gmail.comWe presented a model of large elastocreep deformations. Separation of Almansi strain tensor is determined by the quadratic formof reversible and irreversible components. We consider spherically symmetric deformation of a hollow sphere in the steady creepprocess. Numerical solution of boundary-value problem was obtained. A method for determining loading force on the deformedstate was proposed. Functions of the external loading force according to the laws of a given change in the displacement field wereconstructed.Key words: large deformation, creep, relaxation, elasticity.

Numerical solution of boundary-value problems for quasistatical flow of viscoplastic medium with negative strain rate sensitivity

Numerical solution of boundary-value problems for quasistatical flow of viscoplastic medium with negative strain rate sensitivity

Automatic Fréchet Differentiation for the Numerical Solution of Boundary-Value Problems

A new solver for nonlinear boundary-value problems (BVPs) in Matlab is presented, based on the Chebfun software system for representing functions and operators automatically as numerical objects. The solver implements Newton’s method in function space, where instead of the usual Jacobian matrices, the derivatives involved are Fréchet derivatives. A major novelty of this approach is the application of automatic differentiation (AD) techniques to compute the operator-valued Fréchet derivatives in the continuous context. Other novelties include the use of anonymous functions and numbering of each variable to enable a recursive, delayed evaluation of derivatives with forward mode AD . The AD techniques are applied within a new Chebfun class called which allows users to set up and solve nonlinear BVPs, both scalar and systems of coupled equations, in a few lines of code, using the “nonlinear backslash” operator (\). This framework enables one to study the behaviour of Newton’s method in function space.

Compact difference schemes of high-order accuracy

UDC 519.62 We survey the main results on the construction and study of exact and truncated compact difference schemes of high-order accuracy for a numerical solution of boundary-value problems for ordinary differential equations and the development of new efficient algorithms for the numerical solution of boundary-value problems with given accuracy and automatic selection of the grid points on their basis.

Exact three-point difference scheme for a nonlinear boundary-value problem on the semiaxis

For the numerical solution of boundary-value problems on the semiaxis for second-order nonlinear ordinary differential equations, an exact three-point difference scheme is constructed and substantiated. Under the conditions of existence and uniqueness of solution of a boundary-value problem, we prove the existence and uniqueness of solution of the exact three-point difference scheme and convergence of the method of successive approximations for its solution.