Nonsteady temperature field in thermosensitive body with discontinuous boundary conditions of the second kind

A method of determining the nonsteady temperature fields in bodies with discontinuous boundary conditions is proposed, taking account of the temperature dependence of the thermophysical characteristics.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 53, No. 3, pp. 459–467, September, 1987.     

Heat diffusion with time-dependent convective boundary conditions

The heating (or cooling) of the rock adjacent to water flowing through a crevice is of interest in certain geothermal studies. For many purposes, only the rock temperature at the rock-water interface is needed. The fact that the convective heat transfer coefficient is a function of time in the case of unsteady flow takes this problem out of the strictly classical domain. Several methods of solution are described. An approach in which the heat equation is solved, under appropriate side conditions, by standard finite difference techniques is shown to be satisfactory but rather wasteful because much unneeded information concerning internal rock temperatures must be obtained. Several integral equations of Volterra type are derived that provide the temperature only at the interface. Two numerical approaches to their solution are described: one quite classical and only partially effective; the other an apparently new algorithm that is fast and efficient. Analytical upper and lower bounds on the temperature are obtained and serve to demonstrate that this numerical device is also very accurate. The algorithm is applicable to a wide class of Volterra integral equations of the second kind. A brief discussion of the possible future uses of this scheme and the need for additional research is given.     

A boundary value technique for the solution of some diffusion equations with concentration-dependent diffusion coefficients

A boundary value method is applied to the problem of diffusion equations with concentration-dependent diffusion coefficients. The resulting non-linear system of difference equations is solved with exceptional speed by using a form of the Gauss elimination method. This method of solving problems with a semi-infinite domain was found to be more accurate and up to ten times faster than initial value techniques.     

An analytical method of calculating variable diffusion coefficients

An analytical method of calculating variable diffusion coefficients has been developed. The Boltzmann-Matano graphical method used up to now has certain disadvantages which restrict the acuracy of the evaluated diffusion coefficients. Due to these disadvantages, the accuracy of this method is restricted even more when diffusion coefficients are determined in an area of low concentration of the diffused species of atom. The reliability of the analytical expression obtained in the present investigation has been verified experimentally by comparing the results obtained with those of the graphical method. On the basis of this comparison the analytical method can claim an accuracy better than, or at least equal to, that of the graphical method.     

Temperature patterns in soils with variable thermophysical coefficients with given temperature conditions in greenhouses

An analytic method is proposed for solving nonstationary heat conduction with transport coefficients dependent on the coordinates. The temperature distributions in soils have been examined for harmonic and exponential laws of variation for the air temperature in greenhouses.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 42, No. 2, pp. 313–320, February, 1982.     

A boundary element method for a second order elliptic partial differential equation with variable coefficients

A boundary element method is derived for solving a class of boundary value problems governed by an elliptic second order linear partial differential equation with variable coefficients. Numerical results are given for a specific test problem.     

Prebuckling and buckling analysis of variable angle tow plates with general boundary conditions

The concept of variable angle tow (VAT) placement is explored for enhancing the buckling resistance of composite plates subjected to axial compression loading. The problem is relatively difficult to solve because of varying stiffness properties and requires prior prebuckling analysis to determine the non-uniform stress variation followed by the buckling analysis of VAT plates. A stress function formulation for in-plane analysis and displacement formulation for buckling analysis was employed to derive the governing differential equations of VAT plates based on classical laminated plate theory. The Differential Quadrature Method (DQM) is applied to solve the differential equations. The novel aspect of the present work is the use of Airy’s stress function to model the prebuckling analysis of VAT plates which considerably reduces the problem size, computational effort and provides generality to model pure stress and mixed boundary conditions. DQM was applied first to solve the prebuckling problem of VAT plates subjected to cosine distributed compressive load/ uniform end shortening. Then, DQM was applied to solve the buckling problem of rectangular VAT plates subjected to axial compression under different plate boundary conditions. Results were validated with detailed finite element analysis and the relative accuracy and efficiency of the DQM approach is discussed.     

Transient meshless boundary element method for prediction of chloride diffusion in concrete with time dependent nonlinear coefficients

Chloride-induced corrosion of steel reinforcements has been identified as one of the main causes of deterioration of concrete structures. A feasible numerical method is required to predict chloride penetration in concrete structures. A transient meshless boundary element method is proposed to predict chloride diffusion in concrete with time dependent nonlinear coefficient. Taking Green's function as the weighted function, the weighted residue method is adopted to transform the diffusion equation into equivalent integral equations. By the coupling of radial integral method and radial basis function approximation, the domain integrals in equivalent control equations are transformed into boundary integrals. Following the general procedure of boundary element meshing and traditional finite difference method, a set of nonlinear algebraic equations are constructed and are eventually solved with the modified Newtonian iterative method. Several numerical examples are provided to demonstrate the effectiveness and efficiency of the developed model. A comparison of the simulated chloride concentration with the corresponding reported experimental data in a real marine structure indicates the high accuracy and advantage of the time dependent coefficient and nonlinear model over the conventional constant coefficient model.     

Radial integration boundary integral and integro-differential equation methods for two-dimensional heat conduction problems with variable coefficients

This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional heat conduction problems with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the boundary-value problem (BVP) to a boundary-domain integral equation (BDIE) or boundary-domain integro-differential equation (BDIDE). The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Heat conduction with variable heat transmission coefficients

An analytical method is shown for determining the temperature field of a body whose variable thermophysical properties are functions of the space coordinates. The problem is solved for a plate and for a cylinder where the thermal conductivity is an exponential function of one space coordinate.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 25, No. 2, pp. 353–362, August, 1973.     

An implicit corrected SPH formulation for thermal diffusion with linear free surface boundary conditions

The smoothed particle hydrodynamics (SPH) method has proven useful for modeling large deformation of fluids including fluids with stress‐free surfaces. Because of the Lagrangian nature of the method, it is well suited to address the thermal evolution of these free surface flows. Boundary conditions at the interface of the fluid with a solid wall are usually enforced through the use of boundary particles. However, applying conditions at free surfaces, in particular gradient boundary conditions, can be problematic with traditional SPH formulations due to the degradation of the gradient approximation in these regions. Compounding this difficulty is that traditional approximations of the Laplacian operator suffer a similar degradation near free surfaces. A new SPH formulation of the Laplacian operator is presented, which improves the accuracy near free surface boundaries. This new form is based on a gradient approximation commonly used in thermal, viscous, and pressure projection problems, but includes higher‐order terms in the appropriate Taylor series. Comparisons with other approximations of second‐order derivatives are given. The discretization is tested by solving steady‐state and transient problems of thermal diffusion using the Backward Euler method with a GMRES solver. Boundary conditions are imposed through an augmented matrix. Copyright © 2008 John Wiley & Sons, Ltd.     

计及非齐次边界条件的面内变速运动黏弹性板的稳态响应

研究非齐次边界条件和1∶3内共振下面内平动黏弹性板的横向非线性1∶2主参数振动的稳态响应。考虑黏弹性对边界条件的影响,建立了面内平动板的偏微分运动方程和相应的非齐次边界条件。采用直接多尺度法建立了次谐波参数共振时的可解性条件,并根据Routh-Hurvitz判据判别了系统幅频响应的稳定性。讨论了速度扰动幅值和黏弹性系数对幅频响应的影响,对比了齐次和非齐次边界条件下稳态响应的差异。最后,引入微分求积法验证直接多尺度法的近似解析结果。     

Stability criteria for explicit finite difference solutions of the parabolic diffusion equation with non-linear boundary conditions

The criteria for stability of the explicit finite difference solution of the one-dimensional, transient, conduction heat transfer problem with both radiant and convection heat transfer at the boundaries are considered in this paper. These criteria are governed by an inequality set from a functional relationship between the newly calculated and the old temperature at each node. From the node with the most stringent criteria, it is shown that setting the coefficient of the old temperature equal to zero in the governing difference equation is not sufficient for a general criterion. On the other hand, setting the derivative of the new temperature with respect to the old temperature equal to zero in the governing difference equation presents a simple, straightforward technique for obtaining a sufficient condition for a stable system. It is further shown that the second law of thermodynamics, written in explicit finite difference form, does present a necessary criterion for stability. However, the second law, because it is in the form of an inequality, does not present as simple a criterion as the derivative method does. The specific problem studied is a finite thickness slab, initially at a uniform temperature, but instantaneously subjected to both radiation and convection on its two surfaces. Temperature profiles were calculated on a digital computer and are presented in dimensionless graphical form over a range of five dimensionless parameters. A plot that relates stability to the maximum time-step size for the entire range of practical conditions of radiation numbers is also presented.     

Nonsteady thermodiffusional separation of binary mixtures under sampling conditions

An accurate solution is obtained for the nonlinear nonsteady problem of the thermodiffusional separation of two-component mixtures in sampling conditions.     

Inverse problem in optical diffusion tomography. II. Role of boundary conditions

We consider the inverse problem of reconstructing the absorption and diffusion coefficients of an inhomogeneous highly scattering medium probed by diffuse light. The role of boundary conditions in the derivation of Fourier-Laplace inversion formulas is considered. Boundary conditions of a general mixed type are discussed, with purely absorbing and purely reflecting boundaries obtained as limiting cases. Four different geometries are considered with boundary conditions imposed on a single plane and on two parallel planes and on a cylindrical and on a spherical surface.     

Solute transport modelling with the variable temporally dependent boundary

In this present study, analytical and numerical solutions are obtained for solute transport modelling in homogeneous semi-infinite porous medium. The dispersion coefficient is assumed to be initial dispersion and velocity is assumed to be temporally dependent with initial seepage velocity. Also, the concept of dispersion is directly proportional to the square of the seepage velocity used for finding the solution. Initially, the domain is not solute free. At one end of the domain, source concentration with the effect of different temporally dependent functions taken into account. The concentration gradient assumed to be zero due to no mass flux at other end of the domain. Laplace Transform Technique is used to obtain the exact solution, whereas Explicit Finite Difference method is used for approximate solution. The different types of temporally dependent velocity are used for the graphical representation of the solution. The accuracy of the solution explored by the Relative error analysis.     

Approximate method for the numerical solution of unsteady heat conduction problems with variable boundary conditions of the third kind

An approximate method for solving unsteady heat conduction problems with variable boundary conditions of the third order, which can be used for bodies of complex shape, is described.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 4, pp. 688–696, September, 1969.