A new boundary integral equation for notch problem of antiplane elasticity

In this paper the notch problem of antiplane elasticity is discussed and a new boundary integral equation is formulated. In the problem, the distributed dislocation density is taken to be the unknown function. Unlike the usual choice, the resultant force function is taken as the right hand term of the integral equation; therefore, a new boundary integral equation for the notch problem of antiplane elasticity with a weaker singular kernel (logarithmic) is obtained. After introducing a particular fundamental solution of antiplane elasticity, the notch problem for the half-plane is discussed and the relevant boundary integral equation is formulated. The integral equations derived are compact in form and convenient for computation. Numerical examples demonstrated that high accuracy can be achieved by using the new boundary equation.     

A DRBEM model for harbor oscillation with the effect of energy dissipation

Abstract

In this study, the numerical scheme of dual reciprocity boundary element method (DRBEM) is adopted to investigate the resonant problem in a harbor while considering the effect of energy dissipation. The numerical model employed the mild slope equation as a basic equation. To avoid complicated procedures for solving the equation, DRBEM is used to improve numerical efficiency. Computation results are compared with the existing experimental data and other theoretical results. It shows that the present model is valid and effective to solve the harbor oscillation problem.     

A BEM sensitivity and shape identification analysis for acoustic scattering in fluid–solid problems

In this paper a boundary element formulation for the sensitivity analysis of structures immersed in an inviscide fluid and illuminated by harmonic incident plane waves is presented. Also presented is the sensitivity analysis coupled with an optimization procedure for analyses of flaw identification problems. The formulation developed utilizes the boundary integral equation of the Helmholtz equation for the external problem and the Cauchy–Navier equation for the internal elastic problem. The sensitivities are obtained by the implicit differentiation technique. Examples are presented to demonstrate the accuracy of the proposed formulations. © 1998 John Wiley & Sons, Ltd.     

Heating of massive bodies by variable-intensity radiant heat source

Radiant heating of solid bodies in a variable-temperature medium is investigated. The problem is reduced to a nonlinear Volterra integral equation of the second kind, for which the classical successive-approximation method converges. Subsequent analysis of the resulting functional equation brings to light certain properties of the temperature field; they are used as a basis for an engineering computational method for the given heat-conduction problem.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 2, pp. 292–299, August, 1969.     

Solution of a plane,axisymmetric and three-dimensional single-phase stefan problem

Using a superposition method we construct a solution of the multidimensional problem of the steady-state fusion regime of semibounded solids. The solution of the problem is reduced to a generalized Fredholm integral equation of the first kind. A method is given for solving the integral equation for plane problems by converting to a linear system of algebraic equations.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 27, No. 2, pp. 341–350, August, 1974.     

Nonstationary torsion of a tapered shaft weakened by a spherical crack

We obtain an approximate effective solution of the problem of nonstationary stress concentration near a spherical crack located inside a tapered shaft whose face is subjected to the action of impact tangential stresses. To solve the problem, we propose to use an approach based on the discretization of the problem in time with the help of a difference scheme. By the method of integral transformations, the problem is reduced to an integral equation for the unknown jump of displacements on the crack. This equation is solved by the method of orthogonal polynomials. The relation for the evaluation of the stress intensity factors is obtained. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 1, pp. 49–55, January–February, 2008.     

Modelling of an Oscillatory Magnetic Field Action on a Ferrofluid Layer

We formulate the parametric resonance problem corresponding to an isothermal ferrofluid layer subjected to a magnetic field that consists of two parts. The first one is a vertical or a horizontal constant component. The second part is oscillating with time in a vertical plane. The Laplace law becomes then equivalent to the study of a Hill equation, thus generalizing the Mathieu equation studied for a purely oscillating magnetic field. This extends the classical Faraday vibrating problem to other experimental situations.     

Optimal distribution of a heat source within a thermopenetrator

Heat conduction within a heater of an arbitrary shape is investigated. A mathematical model is presented as a mixed boundary-value problem for the Poisson equation converted into a Fredholm boundary integral equation of the first kind which is solved numerically. A closed-form solution for the particular case of a rectangular heater is also found. Provided that the temperature and heat flux on the heater's boundary are given, the problem is treated as an inverse problem where the heat source distribution within the heater is the unknown function. The existence of the unique solution of this inverse problem is proved. Finally, the problem is solved numerically for a one-dimensional heat source.     

On the Euler equation appearing in the synthesis of radiating systems

A problem appearing when the Euler equation is used in the theory of antenna synthesis is considered. Behavior of a solution to this equation at the ends of the synthesis interval contradicts the Meixner conditions, according to which this solution must tend to zero as a square root of the distance to the ends. It is shown that this contradiction arises if we seek solutions in the L ₂ space and disappears if an L ₂ subspace with limited energy norm is selected as the space of solutions. In this case, the Euler equation arises from a variational minimization problem for a functional involving the norm of current determined in the energy space.     

Cartesian grid methods using radial basis functions for solving Poisson, Helmholtz, and diffusion–convection equations

Four methods that solve the Poisson, Helmholtz, and diffusion–convection problems on Cartesian grid by collocation with radial basis functions are presented. Each problem is split into a problem with an inhomogeneous equation and homogeneous boundary conditions, and a problem with a homogeneous equation and inhomogeneous boundary conditions. The former problem is solved by collocation with multiquadrics, whereas the latter problem is solved by collocation with either multiquadrics or fundamental solutions. It is found that methods that make use of fundamental solutions for collocation yield more accurate solutions that are less sensitive to the shape parameter of multiquadrics and node arrangement. Additional collocation appears to improve the quality of solutions.     

Solution of low péclet number heat transfer with viscous dissipation in the semi‐infinite axial region of a tube via functional analytic methods

Abstract

A solution of the extended Graetz problem with prescribed wall heat flux and viscous dissipation in a semi‐infinite axial region of a tube is obtained by functional analytic methods. The energy equation is split into a set of partial differential equations to obtain a self‐adjoint formulism. Then, an algebraic characteristic equation of the eigenvalue problem for an arbitrary velocity profile is obtained by an approximation method in L ₂[0, 1]. In addition, a backward recursive formula for calculating the expansion coefficients of the solution is developed.     

Numerical examination for degenerate scale problem for ellipse‐shaped ring region in BIE

This paper investigates the degenerate scale problem for an ellipse‐shaped ring region in boundary integral equation (BIE). A homogenous integral equation is introduced. The integral equation is reduced to an algebraic equation after discretization. The critical value for the degenerate scale can be obtained from the vanishing condition of a determinant. It is proved that there are two critical values for the degenerate scale, rather than one. This finding is first proposed in the paper. Two particular problems with known solutions are examined numerically. The loadings applied on the exterior boundary may result in a resultant force in the x‐direction or in the y‐direction. The improper numerical solutions have been found once the real size approaches the critical value. Two techniques for avoiding the improper solutions are suggested. The techniques depend on the appropriate choice of the used size or adding a constant in a kernel of the integral equation. It is proved that both techniques will give accurate numerical results. Numerical examinations for the problem are emphasized in the paper. Copyright © 2007 John Wiley & Sons, Ltd.     

Solution of two conjugate equations

A solution to the conjugacy problem of various types of equation has been found by reduction to an integral equation; the problem arises in the study of heat-and masstransfer and in mechanical or electric phenomena in diverse media. Conditions are found which must be imposed on given functions for which the problem has a classical solution.     

随机LQ控制在投资组合模型及套期保值问题中的应用

本文提出一类随机线性二次最优控制问题,给出了一个新的Ricatti方程,若此方程有解,可得到系统的最优反馈控制;作为其应用,文中首先讨论了连续时间的投资组合模型,得到最优证券组合;最后将其运用于金融中未定权益的套期保值问题,并得到最优套期保值策略。     

A nonlinear eigenvalue problem from thin-film flow

Steady solutions of a fourth-order partial differential equation modeling the spreading of a thin film including the effects of surface shear, gravity, and surface tension are considered. The resulting fourth-order ordinary differential equation is transformed into a canonical third-order ordinary differential equation. When transforming the problem into standard form the position of the contact line becomes an eigenvalue of the physical problem. Asymptotic and numerical solutions of the resulting eigenvalue problem are investigated. The eigenvalue formulation of the steady problem yields a maximum value of the contact angle of 63.4349^?.     

Solution of the Pontriagin–Vitt equation for the moments of time to first passage of the randomly accelerated particle by the finite element method

The Pontriagin–Vitt equation governing the mean of the time of first passage of a randomly accelerated particles has been studied extensively by Franklin and Rodemich.¹ In their paper is presented the analytic solution for the two-sided barrier problem and solutions by several finite difference procedures. This note demonstrates solution of the problem by a Petrov–Galerkin finite element method using upstream weighting functions,² shown to give rapidly convergent results. In addition, the equation is generalized to include higher statistical moments, and solutions for the first few ordinary moments are reported.     

A computational algorithm for catalysts experiencing concentration‐dependent deactivation

Abstract

A computational algorithm is proposed for catalyst pellets or reactors experiencing concentration‐dependent deactivation. In the integration of the deactivation equation in each time interval, the concentration of poison, reactant or/and product is considered to be a constant. The value of concentration is recalculated from the mass balance equation before integrating the deactivation equation. By such an approach, the number of equations is reduced; thus a two‐dimensional problem can be converted to a single‐dimensional one.     

Evaluation of the T-stress in branch crack problem

This paper investigates the T-stress in the branch crack problem. The problem is modeled by a continuous distribution of dislocation along branches, and the relevant singular integral equation is obtained accordingly. After discretization of the singular integral equation, the balance for the number of equations and unknowns is well designed. After the singular integral equation is solved, the equation for evaluating the T-stress is derived. The merit of present study is to provide necessary equation for evaluating T-stress, rather than to provide the integral equation. Many computed results for T-stress under different conditions for branch crack are presented. It is found from the computed results that the interaction for T-stress among branches is complicated.     

Orthotropic piezoelectric patches for control of symmetric laminates via an integral equation

Formulation of the problem for the feedback displacement control of a vibrating laminated plate with orthotropic piezoelectric sensors and actuators is given in terms of an integral equation. The objective is to develop a formulation which facilitates the numerical solution to obtain the eigenfrequencies and eigenfunctions of the piezo-controlled plate. The control is carried out via piezoelectric sensors and actuators which are of orthorhombic crystal class mm² with poling in the z direction. The initial formulation of the problem is given in terms of a differential equation which is the conventional formulation most often used in the literature. The conversion to an integral equation formulation is achieved by introducing an explicit Green’s function. Explicit expressions for the kernel of the integral equation are given and the method of solution using the new formulation is outlined. The solution technique involves approximating the integral equation with an infinite system of linear equations and using a finite number of these equations to obtain the numerical results.     

Analytical and numerical solution of the problem of meteoroid motion in terrestrial atmosphere

A system of equations describing meteoroid motion in the terrestrial atmosphere is considered. It is shown that the system can be reduced to a single second-order differential equation for the height dependence of the mass. Approximate analytical expressions for the solution of the Cauchy problem for this equation are obtained, and conditions of applicability of these solutions are determined. The general case of the problem is solved numerically. Results of mathematical modeling are presented. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 2, pp. 342–348, March–April, 1998.