Simultaneous determination for a space-dependent heat source and the initial data by the MFS

In this paper we propose a numerical algorithm based on the method of fundamental solutions for recovering a space-dependent heat source and the initial data simultaneously in an inverse heat conduction problem. The problem is transformed into a homogeneous backward-type inverse heat conduction problem and a Dirichlet boundary value problem for Poisson's equation. We use an improved method of fundamental solutions to solve the backward-type inverse heat conduction problem and apply the finite element method for solving the well-posed direct problem. The Tikhonov regularization method combined with the generalized cross validation rule for selecting a suitable regularization parameter is applied to obtain a stable regularized solution for the backward-type inverse heat conduction problem. Numerical experiments for four examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed algorithm.     

An analogy between rectilinear flow of plastic solids and Saint-Venant torsion

This paper presents a new analogy between steady rectilinear flow of plastic materials and Saint-Venant torsion of elastic bars. Material behaviour is modelled as rigid/linear-hardening in conjunction with the von Mises flow rule. Focusing attention on steady flow through pipes with simply connected cross sections and perfectly rough walls, we arrive at a Dirichlet-type problem. The mathematical formulation of this problem is analogous to that of Saint-Venant torsion of elastic bars. This analogy provides a number of exact solutions for steady rectilinear flow of plastic materials, through pipes with various cross sections. A few simple examples are discussed with emphasis on the growth of plastic boundary layers induced by wall friction. The influence of stress singularities near corners, at the boundary, is also investigated.     

The influence of thermal relaxation and thermal damping on transient processes with cyclic boundary conditions

Variants of the differential equation of heat conduction in a solid body, which follow from the Fourier and Cattaneo–Vernotte hypotheses and the Lykov equation, are considered. A boundary value problem describing temperature fields in a body (cylinder) upon cyclic heat transfer with cold and hot media is formulated. An analytical solution to the boundary value problem with a hyperbolic differential equation of heat conduction with allowance for thermal relaxation and temperature damping with cyclic boundary conditions of the third kind is given. The thermal transient processes calculated by the classical heat conductance equation and hyperbolic equation of heat conduction on the axis of the cylinder at different values of factors such as the ratio of the thermal damping time to the thermal relaxation time, the duration of cyclic periods, the Fourier relaxation number, and the Biot number are compared. A conclusion is made that the theory of regenerative air heater should be improved by taking into account thermal relaxation and thermal damping in the nozzle and measurements of the thermal relaxation and thermal damping times of the corresponding materials.     

Heat conduction in a solid slab embedded with a pipe of general cross-section: Shape Factor and Shape Optimization

We address the problem of two-dimensional heat conduction in a solid slab embedded with an isothermal, symmetric pipe of general cross-section. Similar formulations have applications in continuum mechanics and electricity. The main objective of this work is to develop a Shape Optimization algorithm that will reveal the optimal shapes of the pipe such that the conduction rate is maximized or minimized. This is achieved by optimizing the Shape Factor. To obtain the Shape Factor we transform the pipe into a strip using the generalized Schwarz-Christoffel transformation, and develop an integral equation of the first kind for the temperature gradient using Fourier transform techniques. The integral equation is solved both numerically and analytically/asymptotically. The fact that the Shape Factor is a monotonic function of the length of the strip suggests a Shape Optimization formulation where the objective function is the length of the strip and the variables of the optimization are the parameters of the generalized Schwarz-Christoffel transformation. Optimal shapes for the problem of minimizing the conduction rate are computed numerically and validated with an analytical solution. Numerical results for maximizing the transport rate are also obtained. The versatility and the robustness of the numerical optimization algorithm offers opportunities for improving the design of similar processes with non-linear equality and inequality constraints.     

A note on uniqueness in coupled thermoplasticity

Summary Criteria are derived for uniqueness in coupled thermoplasticity. A small strain thermoplastic constitutive model is developed within the framework of the 1965 Green and Naghdi [1] formulation. An energy balance equation is also formulated which accommodates thermal conduction: when plastic flow is suppressed, the equation reduces to the correct thermoelastic relation. For prescribed tractions and heat fluxes on the boundary, criteria are sought under which the solution of the boundary value problem, assumed unique at the current time, remains unique for a sufficiently short time interval. The criteria are in the form of several matrix inequalities. Since the various matrices are at most of rank 6, numerical application of the inequalities poses no significant difficulty.     

A general procedure transferring domain integrals onto boundary integrals in BEM

This paper is concerned with transferring to the boundary the domain type integrals occurring in the boundary integral approach applied to boundary value problems with nonzero body force terms. The framework used encloses many interesting engineering applications, e.g. elastostatics, heat conduction and magnetostatics. Beside this are pseudo plastic strains are incorporated due to interaction phenomena with relevant quantities. The proposed method is based upon the homogenization of the governing differential equation by using a particular solution of the inhomogeneous one. Various methods deriving such particular solutions are considered.     

Methods of fundamental solutions for time‐dependent heat conduction problems

A time‐dependent heat conduction problem can be solved by the method of fundamental solutions using the fundamental solution to the modified Helmholtz equation or the fundamental solution to the heat equation. This paper presents solutions using both formulations in terms of initial and boundary conditions. Such formulations enable calculation of errors and variance, which indicates sensitivities of solutions to uncertainties in initial and boundary conditions. Both errors and variance of solutions to three test problems by the two methods of fundamental solutions are used to compare performances of the methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Two-dimensional heat conduction with phase change in a semi-infinite mould

Analytical solution of a 2-dimensional problem of solidification of a superheated liquid in a semi-infinite mould has been studied in this paper. On the boundary, the prescribed temperature is such that the solidification starts simultaneously at all points of the boundary. Results are also given for the 2-dimensional ablation problem. The solution of the heat conduction equation has been obtained in terms of multiple Laplace integrals involving suitable unknown fictitious initial temperatures. These fictitious initial temperatures have interesting physical interpretations. By choosing suitable series expansions for fictitious initial temperatures and moving interface boundary, the unknown quantities can be determined. Solidification thickness has been calculated for short time and effect of parameters on the solidification thickness has been shown with the help of graphs.     

Investigation of transient conduction–radiation heat transfer in a square cavity using combination of LBM and FVM

In this paper, the effect of surface radiation in a square cavity containing an absorbing, emitting and scattering medium with four heated boundaries is investigated, numerically. Lattice Boltzmann method (LBM) is used to solve the energy equation of a transient conduction–radiation heat transfer problem and the radiative heat transfer equation is solved using finite-volume method (FVM). In this work, two different heat flux boundary conditions are considered for the east wall: a uniform and a sinusoidally varying heat flux profile. The results show that as the value of conduction–radiation decreases, the dimensionless temperature in the medium increases. Also, it is clarified that, for an arbitrary value of the conduction–radiation parameter, the temperature decreases with decreasing scattering albedo. It is observed that when the boundaries reflect more, a higher temperature is achieved in the medium and on boundaries.     

Solution of laplace equation by the method of separation of variables

Abstract

The Laplace equation, which is used to describe the problem of two‐dimensional heat conduction with appropriate boundary conditions at steady state, is solved in this work by applying the method of separation of variables. The primary objective of this work involves discussing the effects of the constant value of the separation of variables (p) and the sequential order of substituting boundary conditions on the solution. Without appropriately arranging the sequential order of substituting the boundary conditions, the solution for non‐zero constant values of separation of variables (p) can not be obtained. For a zero value for the constant of the separation of variables, the solution obtained is trivial or does not exist. Solutions in different forms are obtained by using different values for the constant of the separation of variables (p) and for the sequential orders of substituting the boundary conditions.     

Additional boundary conditions in unsteady-state heat conduction problems

Using some additional sought function and boundary conditions, a precise analytical solution of the heat conduction problem for an infinite plate was obtained using the integral heat balance method with symmetric first-order boundary conditions. The additional sought function represents the variation of temperature with time at the center of a plate and, due to an infinite heat propagation velocity described with a parabolic heat conduction equation, changes immediately after application of a first-order boundary condition. Hence, the range of its time and temperature variation completely incorporates the ranges of unsteadystate process times and temperature changes. The additional boundary conditions are such that their fulfilment is equivalent the fulfilment of a differential equation at boundary points. It has been shown that the fulfilment of an equation at boundary points leads to its fulfilment inside the region. The consideration of an additional sought function in the integral heat balance method provide a possibility to confine the solution of an equation in partial derivatives to the integration of an ordinary differential equation, so this method can be applied to the solution of equations, which do not admit the separation of variables (nonlinear, with variable physical properties of a medium, etc.).     

Vibration in mode III of an elastic layer containing a crack perpendicular to the boundary edge

A mathematical formulation of the problem of stresses and displacements in an elastic layer which contains a crack perpendicular to the boundary and subjected to a vibrating stress, in mode III, is developed. The boundary conditions for the case of free loading at the edges of the strip are used to obtain a solution to this problem. The problem is reduced to the solution of a Fredholm integral equation. In Part II the problem describing the case of rigid constraint at the edges of an elastic strip containing a vibrating external crack in mode II is reduced to the Fredholm integral equation of the second kind. In Part III the solution of the problem describing the case of a strip containing a vibrational crack in mode III and laying on a rigid boundary is presented.

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Résumé On développe une formulation mathématique du problème des containtes et déplacements dans une couche élastique comportant une fissure perpendiculaire au bord d'une bande et soumise à une sollicitation vibratoire de mode III. Les conditions aux limites de bord suivantes sont envisagées: (1) sollicitation libre en bordure de bande, (2) bridage sévère des bords de la bande et (3) maintien de la bande dans une fixation rigide. Dans les trois cas, le problème est ramené à la solution d'une intégrale de Fredholm.

     

The method of fundamental solutions for a time-dependent two-dimensional Cauchy heat conduction problem

We investigate an application of the method of fundamental solutions (MFS) to the time-dependent two-dimensional Cauchy heat conduction problem, which is an inverse ill-posed problem. Data in the form of the solution and its normal derivative is given on a part of the boundary and no data is prescribed on the remaining part of the boundary of the solution domain. To generate a numerical approximation we generalize the work for the stationary case in Marin (2011) [23] to the time-dependent setting building on the MFS proposed in Johansson and Lesnic (2008) [15], for the one-dimensional heat conduction problem. We incorporate Tikhonov regularization to obtain stable results. The proposed approach is flexible and can be adjusted rather easily to various solution domains and data. An additional advantage is that the initial data does not need to be known a priori, but can be reconstructed as well.     

Application of Mises yield criterion to rotating solid disk problem

Continuous stress–displacement solution to thin rotating solid disk problem is obtained for elastic–perfectly plastic material. The solution follows the classical approach based on equation of motion, Hooke’s law, yield criterion, and conditions of continuity of stresses and/or displacement at the elastic/plastic boundary. It is shown that both the displacement field derived from the flow rule associated with Mises yield criterion and the stress distributions are continuous at the elastic/plastic boundary in contrast with the discontinuous solution based on Tresca yield criterion and its associated flow rule.     

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.     

Generalization of the solution of the Prandtl problem of compression of a plastic layer by two rough plates

The Prandtl problem of compression of a plastic layer between parallel rigid plates is gneralized to the case when the plastic layer is not orthogonal to the compressive force. A new fundamental solution for a perfectly plastic body, stress tensor, and maximum compressive force are obtained on the basis of an analytical solution of a nonlinear partial differential equation. The classical Prandtl solutions follow from the new solution in the particular case for an orthogonal layer. A graphic interpretation of the established dependences is given.Translated from Problemy Prochnosti, No. 12, pp. 70–74, December, 1991.     

New implementation of MLBIE method for heat conduction analysis in functionally graded materials

The meshless local boundary integral equation (MLBIE) method with an efficient technique to deal with the time variable are presented in this article to analyze the transient heat conduction in continuously nonhomogeneous functionally graded materials (FGMs). In space, the method is based on the local boundary integral equations and the moving least squares (MLS) approximation of the temperature and heat flux. In time, again the MLS approximates the equivalent Volterra integral equation derived from the heat conduction problem. It means that, the MLS is used for approximation in both time and space domains, and we avoid using the finite difference discretization or Laplace transform methods to overcome the time variable. Finally the method leads to a single generalized Sylvester equation rather than some (many) linear systems of equations. The method is computationally attractive, which is shown in couple of numerical examples for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.     

Acoustic diffraction by a rigid annular disk

Summary The subject of this paper is the problem of acoustic diffraction by a perfectly rigid annular disk. The method of solution rests on formulating the problem in terms of an integral equation which embodies the steady state wave equation as well as the boundary conditions. This Fredholm integral equation of the first kind is converted into four simultaneous integral equations of the second kind by using Williams' integral equation technique. These four integral equations are subsequently solved by the standard iterative procedure when the frequency of the incident wave is low and the inner radius of the annulus is small.     

Contact problem for porous elastic strip

This paper is concerned with the static problem about a contact of the rigid punch above a linear porous elastic strip based on a rigid half-plane. By using Fourier transform along the strip, the problem is reduced to an integral equation with respect to contact pressure, whose kernel possesses a logarithmic singularity, like in classical mixed boundary value problems. The kernel contains two dimensionless parameters, related to porosity and relative width of the punch. For arbitrary porosity a numerical co-location technique is developed, that permits analysis of the contact pressure versus physical and geometrical parameters.     

One-dimensional temperature profile prediction in multi-layered rigid pavement systems using a separation of variables method

This paper presents an analytical solution for prediction of the one-dimensional (1D) time-dependent temperature profile in a multi-layered rigid pavement system. Temperature at any depth in a rigid pavement system can be estimated by using the proposed solution with limited input data, such as pavement layer thicknesses, material thermal properties, measured air temperatures and solar radiation intensities. This temperature prediction problem is modelled as a boundary value problem governed by the classic heat conduction equations, and the air temperatures and solar radiation intensities are considered in the surface boundary condition. Interpolatory trigonometric polynomials, based on the discrete least squares approximation method, are used to fit the measured air temperatures and solar radiation intensities during the time period of interest. The solution technique employs the complex variable approach along with the separation of variables method. A FORTRAN program was coded to implement the proposed 1D analytical solution. Field model validation demonstrates that the proposed solution generates reasonable temperature profile in the concrete slab for a four-layered rigid pavement system during two different time periods of the year.