The dual integral equation method for solving the heat conduction equation for an unbounded plate

The solution of the two-dimensional nonstationary heat conduction equation in axially symmetrical cylindrical coordinates for an unbounded plate is determined in this paper. A solution of the problem is given in the form of functional series, for which every term of a series represents an unknown function of a second kind integral equation. A new type of dual integral equations is used to solve a given boundary-value problem with the help of a Laplace transform and separation of variables.     

On the solution of certain integral equations of mixed problems of plate bending theory

The problem of the bending of a Kirchhoff-Love plate in the shape of a strip under the impression of a thin linear rigid inclusion fastened at one of the edges of the plate when the other edge of the plate is rigidly clamped is considered. The problem is reduced by a Fourier integral transform to the solution of a convolution-type integral equation of the first kind in a finite segment with a regular kernel. The exact inversion of the principal part of the corresponding integral operator is constructed in the class of functions with non-integrable singularities on the segment edges. An effective asymptotic solution is given for the integral equation under investigation in this class of functions in the whole range of variation of the characteristic parameter λ. The results obtained are verified numerically. Analogous integral equations were examined in /1, 2/. The mode of investigation is similar to that proposed in /3/.     

Quadruple Trigonometrical Series Equations and Their Application to an Inclusion Problem in the Theory of Elasticity

Closed form solution of quadruple series equations involving cosine kernels has been obtained by reducing the series equations into triple Abel's type integral equations which in turn are reduced to a single integral equation. Making use of finite Hilbert transforms the solution of the single integral equation is obtained in closed form. This solution is used to solve an electrostatic problem. The results of this paper have also been used in a two-dimensional elastostatic problem under anti-plane shear and the effect of rigid line inclusions with thickness on the Griffith cracks has been examined. The expressions for shear stress and stress intensity factor at the tip of the crack are obtained. Finally, some numerical results for the stress intensity factor and shear stress distribution are obtained.     

Summatory and integral equations in periodic diffraction problems for elastic waves at defects in stratified media

We consider the diffraction problem for an elastic wave on a periodic set of defects located at the interface of stratified media. We reduce the mentioned problem to a pair summatory functional equation with respect to coefficients of the expansion of the desired wave by quasiperiodic waves (the Floquet waves). Using the method of integral identities, we reduce the pair equation to a regular infinite system of linear equations. One can solve this system by the truncation method. We prove that the integral identity is the necessary and sufficient condition for the solvability of the auxiliary overspecified problem for a system of equations in a half-plane in the elasticity theory. We obtain integral equations of the second kind which are equivalent to the initial diffraction problem.     

Static stability analysis of a thin plate with a fixed trailing edge in axial subsonic flow: Possio integral equation approach

In this work, the static stability of a thin plate in axial subsonic airflow is studied using the framework of Possio integral equation. Specifically, we consider the cases when the plate’s leading edge is free and the plate’s trailing edge is either pinned or clamped. We formulate the problem under consideration using a partial differential equations (PDE) model and then linearize the model about the free stream velocity, density, and pressure, to enable analytical treatment. Based on the linearized model, we introduce a new derivation of a Possio integral equation that relates the pressure jump along the thin plate to the plate’s downwash. The steady state solution to the Possio equation is then used to account for the aerodynamic loads in the plate steady state governing equation resulting in a singular differential-integral equation which is transformed to a singular integral equation that represents the static aeroelastic equation of the plate. We verify the solvability of the static aeroelastic equation based on the Fredholm alternative for compact operators in Banach spaces and the contraction mapping theorem. By constructing solutions to the static aeroelastic equation and matching the nonzero boundary conditions at the trailing edge with the zero boundary conditions at the leading edge, we obtain characteristic equations for the free-clamped and free-pinned plates. The minimum solutions to the characteristic equations are the divergence speeds which indicate when static instabilities start to occur. We show analytically that free-pinned plates are statically unstable. We also construct, analytically, flow speed intervals that correspond to static stability regions for free-clamped plates. Furthermore, we resort to numerical computations to obtain an explicit formula for the divergence speed of free-clamped plates. Finally, we apply the obtained results on piezoelectric plates and we show that free-clamped piezoelectric plates are statically more stable than conventional free-clamped plates due to the piezoelectric coupling.     

偏微分方程边值反问题的数值方法研究

本文研究了奇异积分方程在反边值问题中的应用问题.利用圆周上的自然积分方程及其反演公式,把Laplace方程的边值反问题转化为一对超奇异积分方程和弱奇异积分方程的组合,通过选取三角插值近似奇异积分的计算并构造相应的配置格式,并使用Tikhonov正则化方法求解所得到的线性方程组.数值实验表明了该方法的有效性.     

Hermite三次样条多小波自然边界元法

针对应用自然边界元方法解上半平面的Laplace方程的Neumann边值问题时存在奇异积分的困难,本文提出了Hermite三次样条多小波自然边界元法.Hermite三次样条多小波具有较短的紧支集、很好的稳定性和显式表达式,而且它们在不同层上的导数还是相互正交的.因此,本文将它与自然边界元法相结合,利用小波伽辽金法离散自然边界积分方程,使自然边界积分方程中的强奇异积分化为弱奇异积分,从而降低了问题的复杂性.文中给出的算例表明了该方法的可行性和有效性.     

Integral equations of the first kind in the theory of oscillating plates

A modified matrix of fundamental solutions is used to derive and solve first-kind integral equations for the problem of high-frequency harmonic oscillations of an infinite elastic plate with a hole when Dirichlet or Neumann conditions are prescribed on the boundary curve.     

有一条裂纹的圆形焊接问题

本文讨论了在带圆孔的无限平面中焊接一个不同材料的带裂纹的近似圆板的问题.该问题化为求解解析函数边值问题然后又转化为求解沿裂纹的奇异积分方程.后者的数字解法也已给出.文末并对Ⅰ型、Ⅱ型情况得出了应力强度因子的公式以及数字结果.     

Finite deflection analysis of elastic plate by the boundary element method

An integral equation formulation for finite deflection analysis of thin elastic plates is presented, based on general nonlinear differential equations which are equivalent to the von Kármán equations and by virtue of generalized Green identities. Boundary element discretization is applied and a relaxation iterative approach is employed to solve the nonlinear plate bending problems. A number of numerical examples are given; the results of computation are compared with the analytical solutions and good agreement is observed. It appears that the approach developed in this paper is effective.     

层状横观各向同性饱和地基上圆板的非轴对称振动

研究了层状横观各向同性饱和地基上弹性圆板的非轴对称振动问题．首先,通过方位角的Fourier变换,将圆柱坐标系下横观各向同性饱和土的三维动力方程转化为一阶常微分方程组,基于径向Hankel变换,建立问题的状态方程,求解状态方程后得到传递矩阵;其次,利用传递矩阵,结合层状饱和地基的边界条件、排水条件及层间接触和连续条件,给出了任意简谐激振力作用下层状横观各向同性饱和地基动力响应的通解;然后,按混合边值问题建立层状饱和地基上弹性圆板非轴对称振动的对偶积分方程,并将对偶积分方程化为易于数值计算的第二类Fredholm积分方程,并给出了算例．     

一维热传导方程移动边界识别的计算方法

构造了一种正则化的积分方程方法来由Cauchy数据确定一维热传导方程的移动边界．在将区域延拓至规则区域后,通过Fourier方法将问题转化为一个第一类Volterra积分方程．然后分别用Lavrentiev正则化方法以及Tikhonov正则化方法将不稳定的第一类Volterra积分方程转化为适定的第二类积分方程,并分别将积分方程转化为常微分方程组,并用Runge—Kutta方法数值求解,以及直接离散来求解．最后通过自由边界上的条件得到数值的移动边界．通过一些数值试验表明此方法是有效可行的,并且给出的方法无需迭代,数值计算较简单．     

A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.     

Free vibration analysis of discrete-continuous functionally graded circular plate via the Neumann series method

The Neumann series method has been used for the first time to solve the boundary value problem of free axisymmetric and nonaxisymmetric vibrations of continuous and discrete-continuous functionally graded circular plate on the basis of the classical plate theory. The equation of motion and the general solution for a functionally graded circular plate with a very complex system of a discrete elements attached, such as concentric ring masses, elastic supports, rotational springs, and damping elements are presented for the first time. The particular continuous solutions to the defined differential equations are obtained as the Neumann power series rapidly, absolutely, and uniformly convergent to the exact eigenfrequencies for any physically justified values of the plate's parameters on the basis of the properties of the obtained closed-form kernels of the Volterra integral equations. The multiparametric nonlinear characteristic equations for plate with classical and nonclassical boundary conditions are defined and numerically solved to obtain the full spectrum of eigenfrequencies in a simple way. The effects of the position and stiffness of ring supports and of singularities as the radii of supports shrink to the center of the plate on the dimensionless eigenfrequencies of homogeneous and functionally graded circular plate with sliding support and elastic constraints are comprehensively studied and presented for the first time. The accuracy of the proposed low-computational-cost method is demonstrated by comparison of the numerical results with those available in the literature.     

On the integral equation method for buckling loads

An initial value method for the integral equation of the column is presented for determining the buckling load of columns. The differential equation of the column is reduced to a Fredholm integral equation. An initial value problem is derived for this integral equation, which is reduced to a set of ordinary differential equations with prescribed initial conditions in order to find the Fredholm resolvent. The singularities of the resolvent occur at the eigenvalues. Integration of the equations proceeds until the integrals become excessively large, indicating that a critical load has been reached. To check this method, numerical results are given for two examples, for which the critical load is well known. One is the Euler load of a simply supported beam, and the other case is the buckling load of a cantilever beam under its own weight. The advantage of this initial value method is that it can be applied easily to solve other nonlinear problems for which the critical loads are unknown. This approach will be illustrated in future papers.     

Method of integral equations in the scalar problem of diffraction on a system consisting of a “soft” and a “hard” screen and an inhomogeneous body

We consider the scalar problem on the diffraction of a plane wave on a system of two screens with boundary conditions of the first and the second kind and a solid inhomogeneous body in the semiclassical setting. The original boundary value problem for the Helmholtz equation is reduced to a system of singular integral equations over the body domain and the screen surfaces. We prove the equivalence of the integral and differential statements of the problem, the solvability of the system of integral equations in Sobolev spaces, and the smoothness of its solutions. To solve the integral equations approximately, we use the Bubnov-Galerkin method; we introduce basis functions on the body and the screens and prove the consistency and convergence of the numerical method.     

On a Class of Coupled Variational Formulations for a Nonlinear Parabolic Equation

51.Introducti0nSince198O)stheoriesandapplicationsofboundaryelementmethods(BEM)orboundaryintegralmethods(BIM)havemadegreatsuccessesfortheparaboliclnit1alboundaryvalueproblems(seeL1-12j),andtheapproachhasbeenappliedtonumericalsolutionsofinitialboundaryva1ueproblemssuccessfully(seeL1-5j'L8j).Thepropertiesofboundaryelementoperatorshavebeenstudiedbyboundaryintegralmethodsbymanyauthors(see.[4j,L6J'[7j'L12J).Theseresultsprovideabasisforconvergencesanderrorestimatesfornumericalapproximationofbou…     

Modified boundary integral formulations for the Helmholtz equation

In this paper we describe some modified regularized boundary integral equations to solve the exterior boundary value problem for the Helmholtz equation with either Dirichlet or Neumann boundary conditions. We formulate combined boundary integral equations which are uniquely solvable for all wave numbers even for Lipschitz boundaries Γ=∂Ω. This approach extends and unifies existing regularized combined boundary integral formulations.     

板弯曲问题的具两组高阶基本解序列的MRM方法

讨论了双参数地基上薄板弯曲问题.利用两组高阶基本解序列,即调和及重调和基本解序列,采用多重替换方法(MRM方法),得到了板弯曲问题的MRM边界积分方程.证明了该方程与边值问题的常规边界积分方程是一致的.因此由常规边界积分方程的误差估计即可得到板弯曲问题MRM方法的收敛性分析.此外该方法还可推广到具多组高阶基本解序列的情形.     

On the variational formulation of a transmission problem for the Biot equations

In this article, a variational formulation for the transmission problem of the fluid–bone interaction is formulated. The formulation is based on a modified Biot system of equations for the cancellous bone together with a boundary integral equation formulation of the pressure in the water. Existence and uniqueness for the weak solution of the interaction problem are established in appropriate Sobolev spaces.