Stable Boundary and Internal Data Reconstruction in Two-Dimensional Anisotropic Heat Conduction Cauchy Problems Using Relaxation Procedures for an Iterative MFS Algorithm

We investigate two algorithms involving the relaxation of either the given boundary temperatures (Dirichlet data) or the prescribed normal heat fluxes (Neumann data) on the over-specified boundary in the case of the iterative algorithm of Kozlov91 applied to Cauchy problems for two-dimensional steady-state anisotropic heat conduction (the Laplace-Beltrami equation). The two mixed, well-posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation (GCV) criterion. The iterative MFS algorithms with relaxation are tested for over-, equally and under-determined Cauchy problems associated with the steady-state anisotropic heat conduction in various two-dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method.     

Relaxation procedures for an iterative MFS algorithm for two-dimensional steady-state isotropic heat conduction Cauchy problems

We investigate two algorithms involving the relaxation of either the given Dirichlet data (boundary temperatures) or the prescribed Neumann data (normal heat fluxes) on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. [26] applied to two-dimensional steady-state heat conduction Cauchy problems, i.e. Cauchy problems for the Laplace equation. The two mixed, well-posed and direct problems corresponding to each iteration of the numerical procedure are solved using a meshless method, namely the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation (GCV) criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the Laplace operator in various two-dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method.     

A note on the boundary value problem arising in free convection in porous media

By similarity transformation and governing equations of free convection on a heated vertical plate embedded in porous medium are reduced to coupled nonlinear equations. The equations are numerically integrated using the boundary conditions at the plate and at ‘infinity’. Assuming that the plate is subjected to a prescribed temperature [1–3] or to a prescribed heat flux [4, 5], the boundary value problems have been solved independently. These researchers seem to have not noted that the solutions for the two cases are dependent on each other. In the present note we consider yet another thermal boundary condition, namely, radiation boundary condition [6] at the plate and show that the solutions for the three cases are dependent and one can pass from one solution to the other easily.     

Influence functions, integral formulas, and explicit solutions for thermoelastic spherical wedges

In this study, new exact Green’s functions and a new exact Green-type integral formula for a boundary value problem (BVP) in thermoelasticity for some spherical wedges with mixed homogeneous mechanical boundary conditions are derived. The thermoelastic displacements are subjected to a heat source applied in the inner points of the spherical wedges and to a mixed non-homogeneous boundary heat conditions. When the thermoelastic Green’s function is derived, the thermoelastic displacements are generated by an inner unit point heat source, described by Dirac’s δ-function. All results are obtained in elementary functions that are formulated in a special theorem. Exact solutions in elementary functions for two particular BVPs of thermoelasticity for spherical wedges also are included. In these particular BVPs, the thermoelastic displacements are subjected to a constant temperature (in the first particular BVP) or to a constant heat source (in the second particular BVP). In both BVPs, the constant temperature or the constant heat source is given on the segment of the radius of the quarter-space. On the boundary half-planes of the quarter-space zero temperature and zero heat flux are prescribed.     

Thermal boundary layers over a shrinking sheet: an analytical solution

In this paper, the heat transfer over a shrinking sheet with mass transfer is studied. The flow is induced by a sheet shrinking with a linear velocity distribution from the slot. The fluid flow solution given by previous researchers is an exact solution of the whole Navier–Stokes equations. By ignoring the viscous dissipation terms, exact analytical solutions of the boundary layer energy equation were obtained for two cases including a prescribed power-law wall temperature case and a prescribed power-law wall heat flux case. The solutions were expressed by Kummer’s function. Closed-form solutions were found and presented for some special parameters. The effects of the Prandtl number, the wall mass transfer parameter, the power index on the wall heat flux, the wall temperature, and the temperature distribution in the fluids were investigated. The heat transfer problem for the algebraically decaying flow over a shrinking sheet was also studied and compared with the exponentially decaying flow profiles. It was found that the heat transfer over a shrinking sheet was significantly different from that of a stretching surface. Interesting and complicated heat transfer characteristics were observed for a positive power index value for both power-law wall temperature and power-law wall heat flux cases. Some solutions involving negative temperature values were observed and these solutions may not physically exist in a real word.     

A boundary method of Trefftz type with approximate trial functions

This paper presents a new numerical technique which belongs to the group of the Trefftz type methods. It differs from the other boundary techniques by the use of approximate solutions of the corresponding partial differential equations (PDE) as trial functions. This permits to extend the field of application of this boundary method to the problem where one cannot find an appropriate set of exact solutions. In particular, the application of the method presented to a general PDE of the elliptic type is considered in the paper. The method has also been found to work well for certain boundary value and initial value problems including PDEs of the fourth order and equations of the non-stationary heat transfer with moving boundaries.     

Axial symmetric stationary heat conduction analysis of non-homogeneous materials by triple-reciprocity boundary element method

The heat conduction problems in homogeneous media can be easily solved by the boundary element method. The spatial variations of heat sources as well as material coefficients gives rise to domain integrals in integral formulations for solution of boundary value problems in functionally gradient materials (FGM), since the fundamental solutions are not available for partial differential equations with variable coefficients, in general. In this paper, we present the development of the triple reciprocity method for solution of axial symmetric stationary heat conduction problems in continuously non-homogeneous media with eliminating the domain integrals. In this method, the spatial variations of domain “sources” are approximated by introducing new potential fields and using higher order fundamental solutions of the Laplace operator.     

Heat transfer in a viscoelastic boundary layer flow through a porous medium

This paper examines the viscoelastic fluid flow and heat transfer characteristics in a saturated porous medium over an impermeable stretching surface with frictional heating and internal heat generation or absorption. The heat transfer analysis has been carried out for two different heating processes, namely (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHF-case). The governing equations for the boundary layer flow problem result similar solutions. For the specified five boundary conditions, it is not possible to solve directly the resulting sixth-order nonlinear ordinary differential equation. For the present incompressible boundary layer flow problem with constant physical parameters, the momentum equation is decoupled from the energy equation. Two closed–form solutions for the momentum equation are obtained and identified the realistic solution of the physical problem. Exact solution for the velocity field and the skin-friction are obtained. Also, the solution for the temperature and the heat transfer characteristics are obtained in terms of Kummers function. Asymptotic results for the temperature function for large Prandtl numbers are presented. The work due to deformation in the energy equation, which is essential and escaped from the attention of researchers while formulating the visco-elastic boundary layer flow problems, is considered. Drastic variation in the values of heat transfer coefficient is observed when the work due to deformation is ignored.The authors would like to thank the reviewers for their valuable comments/ suggestions to improve the clarity of the paper.     

A continuous dependence result for one-dimensional nonlinear dielectrics

A continuous dependence result is derived for statical solutions to a class of onedimensional boundary value problems for nonlinear deformable dielectrics, under the assumption that the free energy function satisfies a prescribed inequality.     

A meshless method for solving 1D time-dependent heat source problem

A novel meshless numerical procedure based on the method of fundamental solutions (MFS) and the heat polynomials is proposed for recovering a time-dependent heat source and the boundary data simultaneously in an inverse heat conduction problem (IHCP). We will transform the problem into a homogeneous IHCP and initial value problems for the first-order ordinary differential equation. An improved method of MFS is used to solve the IHCP and a finite difference method is applied for solving the initial value problems. The advantage of applying the proposed meshless numerical scheme is producing the shape functions which provide the important delta function property to ensure that the essential conditions are fulfilled. Numerical experiments for some examples are provided to show the effectiveness of the proposed algorithm.     

Solution of stationary heat-conduction problems for curvilinear regions using eigenfunction expansions (fundamental solutions)

Formulas for solution of stationary problems of heat conduction in bodies of a curvilinear shape have been obtained in explicit form using eigenfunction expansions; an analogous solution has been constructed for temperature fluctuations. An algorithm of computation of the boundary functions for classical regions has been proposed; these functions make it possible to reduce the boundary conditions of the problem to a homogeneous form. The exact fundamental solutions in the region of a rectangle with arbitrary smooth boundary conditions of the 1st kind have been constructed using them. These solutions are fundamental, since they can be used when boundary-value problems and inverse problems with unknown boundary conditions are considered for a wide range of curvilinear regions. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 82, No. 1, pp. 163–169, January–February, 2009.     

Three-dimensional fundamental solutions for transient heat transfer by conduction in an unbounded medium,half-space,slab and layered media

This paper presents analytical Green's functions for the transient heat transfer phenomena by conduction, for an unbounded medium, half-space, slab and layered formation when subjected to a point heat source. The transient heat responses generated by a spherical heat source are computed as Bessel integrals, following the transformations proposed by Sommerfeld [Sommerfeld A. Mechanics of deformable bodies. New York: Academic Press; 1950; Ewing WM, Jardetzky WS, Press F. Elastic waves in layered media. New York: McGraw-Hill; 1957]. The integrals can be modelled as discrete summations, assuming a set of sources equally spaced along the vertical direction. The expressions presented here allow the heat field inside a layered formation to be computed without fully discretizing the interior domain or boundary interfaces.The final Green's functions describe the conduction phenomenon throughout the domain, for a half-space and a slab. They can be expressed as the sum of the heat source and the surface terms. The surface terms need to satisfy the boundary conditions at the surfaces, which can be of two types: null normal fluxes or null temperatures. The Green's functions for a layered formation are obtained by adding the heat source terms and a set of surface terms, generated within each solid layer and at each interface. These surface terms are defined so as to guarantee the required boundary conditions, which are: continuity of temperatures and normal heat fluxes between layers.This formulation is verified by comparing the frequency responses obtained from the proposed approach with those where a double-space Fourier transformation along the horizontal directions [Tadeu A, António J, Simões N. 2.5D Green's functions in the frequency domain for heat conduction problems in unbounded, half-space, slab and layered media. CMES: Computer Model Eng Sci 2004;6(1):43–58] is used. In addition, time domain solutions were compared with the analytical solutions that are known for the case of an unbounded medium, a half-space and a slab.     

Extremum principles for some nonlinear heat transfer problems

Summary Recent results on extremum principles for various nonlinear boundary value problems are applied to heat transfer problems involving space radiators such as fins and other parts of spacecraft. The results are illustrated by obtaining quite accurate variational solutions for such problems involving the fourth-power law of radiation.     

Solution of the first and second boundary-value problems of nonstationary heat conduction for a triangular region

Exact solutions of nonstationary problems of heat conduction are constructed in explicit form for a regular triangle of height h with Dirichlet and Neumann’s boundary conditions and an arbitrary initial condition having the property of triple symmetry in the region of the triangle. These very solutions remain valid also for the region of a rectangular triangle with an acute angle π/6, when there are no heat fluxes on the hypotenuse and smaller side, whereas Dirichlet or Neumann boundary conditions are prescribed on the larger side. Here, symmetry limitations are not imposed on the initial conditions. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 73, No. 5, pp. 911–917, September–October, 2000.     

Heat generation in plane strain compression of a thin rigid plastic layer

This paper presents a theoretical investigation into heat generation in the continued quasi-static plane strain compression of a thin metal strip between two rigid, parallel perfectly rough dies. The strip material is rigid perfectly plastic. The length of the dies is supposed to be much larger than the current strip thickness. The plastic work rate approaches infinity in the vicinity of perfectly rough friction surfaces. Since the plastic work rate is involved in the heat conduction equation, this significantly adds to the difficulties of solutions of this equation. In particular, commercial finite element packages are not capable of solving such boundary value problems. The present approximate solution is given in Lagrangian coordinates. In this case, the original initial/boundary value problem reduces to the standard second initial/boundary value problem for the nonhomogeneous heat conduction equation. Therefore, the Green’s function is available in the literature. An example is presented to illustrate the general solution.     

New Poisson's integral formulas for thermoelastic half-space and other canonical domains

In this paper the functions of influence of unit point heat source onto displacements and Poisson-type integral formula for a boundary value problem (BVP) in thermoelastic half-space, free of loadings on the boundary plane are presented in closed form. The thermoelastic displacements are generated by heat source applied at the inner point of the half-space and by heat flux, prescribed on its boundary. All these results are formulated in a special theorem. Furthermore, the advantages and usefulness of the obtained results are also discussed. The main difficulties to obtain such kind of results are to derive the functions of influence of a unit concentrated force onto elastic volume dilatation Θ^(k) and Green's functions in heat conduction G. For canonical Cartesian domains, these difficulties were addressed successfully, and the above-mentioned functions were derived and published earlier. Thus, it can be presumed that for the Cartesian domains, this paper will open a great possibility to derive new thermoelastic influence functions and Poisson's integral formulas in closed form. Moreover, the technique proposed here will also work for any orthogonal canonical domain, as soon as the lists of functions G and Θ^(k) are completed.     

Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution

This paper is concerned with the development of a numerical procedure for solving complex boundary value problems in plane elastostatics. This procedure—the displacement discontinuity method—consists simply of placing N displacement discontinuities of unknown magnitude along the boundaries of the region to be analyzed, then setting up and solving a system of algebraic equations to find the discontinuity values that produce prescribed boundary tractions or displacements. The displacement discontinuity method is in some respects similar to integral equation or ‘influence function’ techniques, and contrasts with finite difference and finite element procedures in that approximations are made only on the boundary contours, and not in the field. The method is illustrated by comparing computed results with the analytical solutions of two boundary value problems: a circular disc subjected to diametral compression, and a circular hole in an infinite plate under a uniaxial stress field. In both cases the numerical results are in excellent agreement with the exact solutions.     

Heat conduction in an inhomogeneous half-space subject to a nonlinear boundary condition. Application of bergman-type series

Bergman-type series solutions involving iterated complementary error integrals are constructed for nonlinear boundary value problems for heat conduction in an inhomogeneous half-space. In particular, a small-time solution is developed when the nonlinear boundary condition is of the Stefan-Boltzmann type.     

A new integral identity for potential polygonal domain problems described by parametric linear functions

The paper presents a modification of the classical integral identity for two-dimensional potential boundary value problems with a linear segments boundary. The modification consists of describing integral contours in the integral identity by means of parametric linear functions. As a result of the modification, it was possible to obtain a new integral identity, which included the geometry of the boundary in its subinterval functions. The identity can be used for finding solutions in polygonal domains under the condition that the solutions previously obtained on the boundary are arrived at by the new parametric integral equation system. The proposed method makes it possible to obtain solutions of domain problems with no need for the discretization of the boundary geometry. The effectiveness of the method is illustrated by three testing examples.     

Three-dimensional steady-state general solution for transversely isotropic hygrothermopiezoelectric media and its application in fracture

The potential theory method is utilized to derive the steady-state, general solution for three-dimensional (3D) transversely isotropic, hygrothermopiezoelectric media in the present paper. Two displacement functions are introduced to simplify the governing equations. Employing the differential operator theory and superposition principle, all physical quantities can be expressed in terms of two functions, one satisfies a quasi-harmonic equation and the other satisfies a tenth-order partial differential equation. The obtained general solutions are in a very simple form and convenient to use in boundary value problems. As one example, the 3D fundamental solutions are presented for a steady point moisture source combined with a steady point heat source in the interior of an infinite, transversely isotropic, hygrothermopiezoelectric body. As another example, a flat crack embedded in an infinite, hygrothermopiezoelectric medium is investigated subjected to symmetric mechanical, electric, moisture and temperature loads on the crack faces. Specifically, for a penny-shaped crack under uniform combined loads, complete and exact solutions are given in terms of elementary functions, which serve as a benchmark for different kinds of numerical codes and approximate solutions.