Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.     

On the approximate solution of some integral equations of the theory of elasticity and mathematical physics PMM vol. 31, no. 6, 1967, pp. 1117–1131

In this paper the development of the method presented in [1] is carried out with application to two types of integral equations encountered in mathematical physics in the investigation of many mixed problems with circular separation line of boundary conditions and in the investigation of plane mixed problems.

The algorithm is given for reducing these integral equations to solution of equivalent infinite linear algebraic systems. It is proved that the resulting infinite systems are quasi completely regular for sufficiently large values of dimensionless parameter A which enters into the systems. It is shown that reduction (truncation) of infinite systems results in finite systems of linear algebraic equations with almost triangular matrices. The last circumstance simplifies considerably the solution of these finite systems after which the solution of initial integral equations is found from simple equations. For given accuracy of the approximate solution and decrease of parameter λ the number of equations in reduced systems increases.

As an example the solution is presented for the axisymmetric problem of a die acting on an elastic layer lying without friction on a rigid foundation.     

Method of successive approximations for three-dimensional laminar boundary layer problems (locally self-similar case) : PMM vol. 37, n6, 1973, pp. 974–983

We present an analytical method for the computation of problems of incompressible boundary layer theory based on an application of the method of successive approximations. The system of equations is reduced to a form suitable for integration. Parameters characterizing the external flow and the body geometry are contained only in the coefficients of the system and do not enter into the boundary conditions. The transformed momentum equations are integrated across the boundary layer from a current value to infinity with the boundary conditions taken into account. If the integration is made from zero to infinity, then the equations pass over into the Kármán relations. Integrating the system of equations a second time, using the boundary conditions at the wall, we obtain a system of nonlinear integro-differential equations. To solve this system of equations we apply the method of successive approximations. To satisfy the boundary Conditions at infinity we introduce, at each step of the iterations, unknown “governing” functions. From the conditions at the outer side of the boundary layer we obtain additional equations for their determination. With the iterational algorithm formulated in this way, the boundary conditions, both on the body and at the outer side of the boundary layer; are satisfied automatically.We consider a locally self-similar approximation. In this case, relative to the “governing” functions, we obtain an algebraic system of equations. We write out the solution in the first approximation. The results obtained in the first approximation are compared with the results of finite-difference computations for a wide range of problems. The results obtained in this paper are compared with those obtained in [1] for the flow in the neighborhood of a stagnation point. An indication is given of the nonuniqueness of the solutions of the three-dimensional boundary layer equations.     

Convergence of the Neumann Series in BEM for the Neumann Problem of the Stokes System

A weak solution of the Neumann problem for the Stokes system in Sobolev space is studied in a bounded Lipschitz domain with connected boundary. A solution is looked for in the form of a hydrodynamical single layer potential. It leads to an integral equation on the boundary of the domain. Necessary and sufficient conditions for the solvability of the problem are given. Moreover, it is shown that we can obtain a solution of this integral equation using the successive approximation method. Then the consequences for the direct boundary integral equation method are treated. A solution of the Neumann problem for the Stokes system is the sum of the hydrodynamical single layer potential corresponding to the boundary condition and the hydrodynamical double layer potential corresponding to the trace of the velocity part of the solution. Using boundary behavior of potentials we get an integral equation on the boundary of the domain where the trace of the velocity part of the solution is unknown. It is shown that we can obtain a solution of this integral equation using the successive approximation method.     

Mixed problems of axisymmetric elasticity theory for some bodies of revolution

We consider the problem of axisymmetric elasticity theory for a space with an elongated ellipsoidal cavity with mixed boundary conditions of smooth contact on the cavity surface and the main mixed problem of axisymmetric elasticity theory for a hyperboloidal layer formed by the two surfaces of a two-cavity hyperboloid of revolution symmetrical about the plane z = O. The problems are solved by the method of p-analytical functions. The solution of the first problem is reduced to solving a Fredholm integral equation of the second kind. We investigate the behavior of the normal stress near the boundary lines. The solution of the second problem is reduced to solving a system of two Fredholm integral equations of the second kind. Existence and uniqueness of the solution is proved for this system.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 88–101, 1989.     

双参数地基上板弯曲问题的边界积分方程

本文应用广义函数的Fourrier积分变换导出了双参数地基上板弯曲问题的基本解,并将基本解展成一致收敛的级数形式.在此基础上,应用广义Rayleigh-Green公式建立了适用于任意形状、任意边界条件情形的两个边界积分方程,为边界元法在这一问题中的应用提供了理论基础.     

Boundary integral equations for mixed Dirichlet,Neumann and transmission problems

We consider a Helmholtz equation in a number of Lipschitz domains in n ≥ 2 dimensions, on the boundaries of which Dirichlet, Neumann and transmission conditions are imposed. For this problem an equivalent system of boundary integral equations is derived which directly yields the Cauchy data of the solutions. The operator of this system is proved to be injective and strongly elliptic, hence it is also bijective and the original problem has a unique solution. For two examples (a mixed Dirichlet and transmission problem and the transmission problem for four quadrants in the plane) the boundary integral operators and the treatment of the compatibility conditions are described.     

A numerical study of parabolic problems with nonlinear boundary conditions

In this paper, we consider an initial‐boundary value problem for a parabolic equation with nonlinear boundary conditions. The solution to the problem can be expressed as a convolution integral of a Green's function and two unknown functions. We change the problem to a system of two nonlinear Volterra integral equations of convolution type. By using an explicit procedure on the basis of Sinc‐function properties, the resulting integral equations are replaced by a system of nonlinear algebraic equations, whose solution yields an accurate approximate solution to the parabolic problem. Some examples are considered to illustrate the ability of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

On the existence of a generalized solution of a nonlinear evolution system of equations in a domain unbounded in time

We consider a mixed problem with Dirichlet homogeneous boundary conditions and nonzero initial conditions for a nonlinear coupled evolution system of equations in a domain unbounded in time. The conditions of existence of a generalized solution are obtained. It is shown that no solution of the problem exists at a negative initial value of the energy integral.     

屈曲特征值问题的边界元方法及收敛性分析

讨论了屈曲特征值问题的定解条件,建立了相应的具约束的积分方程组及带Lagrange乘子的边界变分方程,给出了解的存在唯一性定理。建立了相应的边界元方法并讨论了近似解的误差估计。文末给出了数值算例。     

Application of similarity to research of new boundary integral equations for plate flexure problems

Plate flexure problems are described by a system of boundary integral equations. It is shown that simpler equations can be obtained by using the properties of similarity between two plates. The exact solution of some problems pertaining to circular plates with a concentrated load is given. A sample problem is solved numerically to illustrate the potential of this new formulation.     

含有积分、反周期和带P-Laplacian算子的分数阶微分方程边值问题解的存在性

利用格林函数的性质和Banach压缩映射原理讨论了含P-Laplacian算子反周期边值问题的解.首先,求出与该边值问题相关的格林函数并给出了格林函数的性质;然后将边值问题转化为与其等价的积分方程,利用格林函数的性质及Banach压缩映射原理得到边值问题解的唯一性;最后给出实例验证结果的合理性.     

On the solvability of a boundary value problem for the Laplace equation on a screen with a boundary condition for a directional derivative

We consider a three-dimensional boundary value problem for the Laplace equation on a thin plane screen with boundary conditions for the “directional derivative”: boundary conditions for the derivative of the unknown function in the directions of vector fields defined on the screen surface are posed on each side of the screen. We study the case in which the direction of these vector fields is close to the direction of the normal to the screen surface. This problem can be reduced to a system of two boundary integral equations with singular and hypersingular integrals treated in the sense of the Hadamard finite value. The resulting integral equations are characterized by the presence of integral-free terms that contain the surface gradient of one of the unknown functions. We prove the unique solvability of this system of integral equations and the existence of a solution of the considered boundary value problem and its uniqueness under certain assumptions.     

Vibration op two circular stamps in a layered medium : PMM vol. 40, n 6, 1976, pp. 1104–1112

The problem of the harmonic oscillations on the surface of an elastic layered medium by two circular stamps of radii a₁ and a₂ is considered. The spacing between the centers of the stamps is b > a₁ + a₂. It is assumed that there is no friction in the area of contact.By using a method developed in [1], the initial system is reduced to a system of Fredholm equations of the second kind, for whose solution approximate methods are proposed. On the basis of the results obtained, an applied theory for the vibrations of two stamps can be constructed which also takes account of the dispersion properties of the medium, in contrast to all other known applied theories.It is simple to investigate the case of vibrations of a system of n stamps by the method elucidated in this paper, however, we limit ourselves to the case of two stamps for the sake of brevity.     

The Love's problem for hyperbolic thermoelasticity

In our paper we investigated the initial-boundary value problem for elastic layer situated on half space of another elastic medium. In this medium the thermomechanical interactions were taken into consideration. The system of equations with initial-boundary conditions describes the phenomenon of wave propagation with finite speed. In our problem there are two surfaces ie. free surface and contact surface between layer and half space. On the free surface are setting boundary conditions for normal and tangent surface force. We consider two types of contact between layer and half-space: rigid contact and slip contact. The initial-boundary value problem was solved by using integral transformations and Cagniard-de Hoope methods. From the solution of this problem follows that in layer and half space exist some kind of thermoelastic waves. We investigated moreover the conditions which should be fullfiled for propagation of Rayleigh and Love's type waves on the contact surface between layers and half space. The results obtained in our investigation were used in technical applications especially engineering design and diagnostics of roads and airfields. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A boundary integral equation procedure for the mixed boundary value problem of the vector helmholtz equation

A boundary integral method is developed for the mixed boundary value problem for the vector Helmholtz equation in R³. The obtained boundary integral equations for the unknown Cauchy data build a strong elliptic system of pseudodifferential equations which can therefore be used for numerical computations using Galerkin's procedure. We show existence, uniqueness and regularity of the solution of the integral equations. Especially we give the local "edge" behavior of the solution near the submanifold which divides the Dirichlet boundary from the Neumann boundary     

Pasternak地基上简支板振动问题的准格林函数方法

提出一种新的数值方法——准格林函数方法．以Pasternak地基上简支多边形薄板的振动问题为例,详细阐明了准格林函数方法的思想．即利用问题的基本解和边界方程构造一个准格林函数,这个函数满足了问题的齐次边界条件,采用格林公式将Pasternak地基上薄板自由振动问题的振型控制微分方程化为两个耦合的第二类Fredholm积分方程．边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程来表示问题的边界,以克服积分核的奇异性,最后由积分方程的离散化方程组有非平凡解的条件,求得固有频率．数值方法表明,该方法具有较高的精度．     

The first initial boundary value problem for a compound type equation in a three-dimensional multiply connected region

The method of boundary integral equations is used for solving the first initial boundary value problem for a compound type equation in a three-dimensional multiply connected region. The problem is reduced to a uniquely solvable integral equation. The solution of the problem is obtained in the form of dynamic potentials whose density satisfies this integral equation. Thus the existence theorem is proved. Moreover, the uniqueness of the solution is also studied. All the results are valid for both interior and exterior regions provided that the corresponding conditions at infinity are taken into account. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 249–265, August, 2000.     

Boundary elements solution of stokes flow between curved surfaces with nonlinear slip boundary condition

This work presents a boundary integral equation formulation for Stokes nonlinear slip flows based on the normal and tangential projection of the Green's integral representational formulae for the velocity field. By imposing the surface tangential velocity discontinuity (slip velocity) in terms of the nonlinear slip flow boundary condition, a system of nonlinear boundary integral equations for the unknown normal and tangential components of the surface traction is obtained. The Boundary Element Method is used to solve the resulting system of integral equations using a direct Picard iteration scheme to deal with the resulting nonlinear terms. The formulation is used to study flows between curved rotating geometries: i.e., concentric and eccentric Couette flows and single rotor mixers, under nonlinear slip boundary conditions. The numerical results obtained for the concentric Couette flow is validated again a semianalytical solution of the problem, showing excellent agreements. The other two cases, eccentric Couette and single rotor mixers, are considered to study the effect of different nonlinear slip conditions in these flow configurations. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

The behavior of plasma with an arbitrary degree of degeneracy of electron gas in the conductive layer

We obtain an analytic solution of the boundary problem for the behavior (fluctuations) of an electron plasma with an arbitrary degree of degeneracy of the electron gas in the conductive layer in an external electric field. We use the kinetic Vlasov–Boltzmann equation with the Bhatnagar–Gross–Krook collision integral and the Maxwell equation for the electric field. We use the mirror boundary conditions for the reflections of electrons from the layer boundary. The boundary problem reduces to a one-dimensional problem with a single velocity. For this, we use the method of consecutive approximations, linearization of the equations with respect to the absolute distribution of the Fermi–Dirac electrons, and the conservation law for the number of particles. Separation of variables then helps reduce the problem equations to a characteristic system of equations. In the space of generalized functions, we find the eigensolutions of the initial system, which correspond to the continuous spectrum (Van Kampen mode). Solving the dispersion equation, we then find the eigensolutions corresponding to the adjoint and discrete spectra (Drude and Debye modes). We then construct the general solution of the boundary problem by decomposing it into the eigensolutions. The coefficients of the decomposition are given by the boundary conditions. This allows obtaining the decompositions of the distribution function and the electric field in explicit form.