Heat Diffusion Problem Research Articles

Group method solution for solving nonlinear heat diffusion problems

The linear transformation group approach is developed to simulate heat diffusion problems in a media with the thermal conductivity and the heat capacity are nonlinear and obeyed a striking power law relation, subject to nonlinear boundary conditions due to radiation exchange at the interface according to the fourth power law. The application of a one-parameter transformation group reduces the number of independent variables by one so that the governing partial differential equation with the boundary conditions reduces to an ordinary differential equation with appropriate corresponding conditions. The Runge–Kutta shooting method is used to solve the nonlinear ordinary differential equation. Different parametric studies are worked out and plotted to study the effect of heat transfer coefficient, density and radiation number on the surface temperature.

Numerical solution of a heat diffusion problem by boundary element methods using the Laplace transform

This paper is concerned with a heat diffusion problem in a half-space which is motivated by the detection of material defects using thermal measurements. This problem is solved by inverting the Laplace transform with respect to time on a contour in the complex plane using an exponentially convergent quadrature rule. This leads to a finite number of time-independent problems, which can be solved in parallel using boundary integral equation methods. We provide a full numerical analysis of this scheme on compact time intervals. Our results are formulated in a way that they can easily be used for other diffusion problems in exterior or interior domains.

Semi-analytical method for solving nonlinear heat diffusion problems in spherical medium

A semi-analytical methodology, based on the finite integral transform technique, is proposed to solve the heat diffusion problem in a spherical medium subject to nonlinear boundary conditions due to radiation exchange at the interface according to the fourth power law. The method proceeds by treating the nonlinearity term in the boundary condition as a source in the differential equation and keeping other conditions unchanged. The results obtained from this semi-analytical solutions are compared with those obtained from a numerical solution developed using an explicit finite difference method, which showed very good agreement.

Self-normalized photoacoustic methodology for thermal diffusivity measurements of transparent polymer foils

The analytical solution for the one-dimensional heat diffusion problem for a two-layer system, in the surface absorption model, is used to provide of a self-normalized photoacoustic methodology for thermal diffusivity measurements of transparent polymer foils. The implementation of this method involving linear fits requires one face of the sample to be opaque by means of a thin metal layer. Two different materials were examined using the proposed methodology: acetate and PVDF polymer foils. Good agreement with values reported in the literature was found.

An Approximate Method for Solving Nonlinear Heat Diffusion Problems in Spherical Media

An Approximate Method for Solving Nonlinear Heat Diffusion Problems in Spherical Media

A fast multi‐level convolution boundary element method for transient diffusion problems

AbstractA new algorithm is developed to evaluate the time convolution integrals that are associated with boundary element methods (BEM) for transient diffusion. This approach, which is based upon the multi‐level multi‐integration concepts of Brandt and Lubrecht, provides a fast, accurate and memory efficient time domain method for this entire class of problems. Conventional BEM approaches result in operation counts of order O(N2) for the discrete time convolution over N time steps. Here we focus on the formulation for linear problems of transient heat diffusion and demonstrate reduced computational complexity to order O(N3/2) for three two‐dimensional model problems using the multi‐level convolution BEM. Memory requirements are also significantly reduced, while maintaining the same level of accuracy as the conventional time domain BEM approach. Copyright © 2005 John Wiley & Sons, Ltd.

A multi-level boundary element method for Stokes flows in irregular two-dimensional domains

Recently, we have developed multi-level boundary element methods (MLBEM) for the solution of the Laplace and Helmholtz equations that involve asymptotically decaying non-oscillatory and oscillatory singular kernels, respectively. The accuracy and efficiency of the fast boundary element methods for steady-state heat diffusion and acoustics problems have been investigated for square domains. The current work extends the MLBEM methodology to the solution of Stokes equation in more complex two-dimensional domains. The performance of the fast boundary element method for the Stokes flows is first investigated for a model problem in a unit square. Then, we consider an example problem possessing an analytical solution in a rectangular domain with 5:1 aspect ratio, and finally, we study the performance of the MLBEM algorithm in a C-shaped domain.

A MULTILEVEL BOUNDARY-ELEMENT METHOD FOR TWO-DIMENSIONAL STEADY HEAT DIFFUSION

A fast, accurate, and efficient multilevel boundary-element method (MLBEM) is developed to solve general boundary-value problems arising in computational mechanics. Here we concentrate on problems of two-dimensional steady potential flow and present a fast, direct boundary-element formulation. This novel method extends the pioneering work of Brandt and Lubrecht on multilevel multi-integration (MLMI) in several important ways to address problems with mixed boundary conditions. We utilize bi-conjugate gradient methods (BCGMs) and implement the MLMI approach for fast matrix and matrix transpose multiplication for every iteration loop. After introducing a C-cycle multigrid algorithm, we find that the number of iterations for the bi-conjugate gradient methods is independent of the boundary-element mesh discretization for a broad range of steady-state heat diffusion problems. Here, for a model problem in an L-shaped domain, we demonstrate that the computational complexity of the proposed method approaches the desired goal of N ln N, where N is the number of degrees of freedom. For this problem, we show that the MLBEM algorithm reduces computer run times by a factor of 6,295, whereas the memory requirements are reduced 432 times compared to conventional boundary-element methods, while preserving the accuracy of the numerical solution. Furthermore, the method can be extended in a straightforward manner to the solution of many problems in science and engineering that result in very large sets of matrix equations when the associated integral equations are discretized.

Reduction and optimisation of thermal models using Kirchhoff network theory

The network topology reduction is based on an exact mathematical tool. The principle is a reduction of the problem matrix to a few lines and columns. The original reduction method developed in this study is an extension of this technique implemented to transient linear heat diffusion problems. The finite volume model is hence reduced to a few chosen nodes and a new type of element appears, the compensation capacitance. This element, discussed in [Lagonotte et al., European Phys. J. Appl. Phys. 13 (3) (2001) 177], links every node of the model and improves the dynamic behaviour of the model. Two steps of optimisation, concerning the choice of the nodes and the value of the compensation capacitors, are discussed. The method is implemented to an industrial object, a 5.5 kW squirrel cage induction machine, which model is reduced from 1140 down to 10 nodes.

Self-consistent photoacoustic techniques for thermal diffusivity measurements of self-curable resins

Two self-consistent photoacoustic methodologies, for measuring thermal diffusivity of self-curable resins, are presented. These methodologies make use of the analytical solution for the one-dimensional heat diffusion problem for a three-layer system, assuming the surface absorption model. The analytical procedure relies on the normalization of the theoretical photoacoustic signal from the three layers with the corresponding signal from the uppermost two layers, involving linear fits on the normalized amplitude (in semilog scale) and phase, as functions of the modulation frequency root square. The normalization procedure eliminates the frequency-dependent instrumental electronic contribution (transfer function) and some thermophysically nonrelevant proportionality factors in the theoretical equations, thus making the analysis feasible. The thermal diffusivity of two different self-curable resins was measured and good agreement was obtained between results from both the amplitude and phase, showing consistency between these two photoacoustic methodologies.

Normalized photoacoustic techniques for thermal diffusivity measurements of buried layers in multilayered systems

The one-dimensional heat diffusion problem for a three-layer system is solved assuming the surface absorption model. The analytical solution is shown to be suitable for the implementation of normalized depth-profilometric photoacoustic methodologies involving the open photoacoustic-cell configuration for thermal diffusivity measurements in buried underlayers within a three-layer stack. Our normalization procedures eliminate the frequency-dependent instrumental electronic contribution (transfer function) and some thermophysically nonrelevant proportionality factors in the theoretical equations, thus making the depth-profilometric analysis feasible. The measurement methodology is achieved by normalizing the theoretical photoacoustic signal from the three layers with the corresponding signal from the uppermost two layers, involving linear fits to measure the thermal diffusivity of the third underlayer. Three different multilayered materials were examined using the proposed methodologies. High reproducibility of the thermal diffusivity measurements and good agreement with values reported in literature were found. Besides the foregoing procedures, a lumped photoacoustic model was developed, which yields the effective thermal-diffusivity value of the multilayer stack.

Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics. Progress in Nonlinear Differential Equations and Their Applications, Vol 48

7R42. Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics. Progress in Nonlinear Differential Equations and Their Applications, Vol 48. - SN Antontsev (Dept de Matematica, Univ de Beira Interior, Covilha, 6201-001, Portugal), JI Diaz (Dept de Matematica Aplicada, Univ Complutenese, Madrid, 28040, Spain), S Shmarev (Dept de Matematicas, Univ de Oviedo, Oviedo, 33007, Spain). Birkhauser Boston, Cambridge MA. 2002. 329 pp. ISBN 0-8176-4123-8. $79.95.Reviewed by AJ Kassab (Dept of Mech, Mat, and Aerospace Eng, Col of Eng, Univ of Central Florida, Orlando FL 32816-2450).This is a research monograph reporting on collaborative research carried out by the authors over the past 15 years in the field of free boundary problem. Specifically, the authors use energy methods as the basic tool to investigate the underlying mathematical structure of the partial differential equations (PDE) governing various field problems whose solutions lead to the formation of a free boundary. Given the definition of an energy norm, the technique results in inequalities involving ordinary differential equations whose study allows determination of the existence of a free boundary and provides a theoretical framework for the study of the solutions for the PDEs of interest. This book mainly considers application of energy methods to quasi-nonlinear equations of fluid mechanics. The first three chapters lay the formal mathematical and theoretical framework of energy methods applied to nonlinear PDEs. The first chapter introduces energy methods to study nonlinear stationary problems giving rise to localized solutions and hence free boundaries. Second-order elliptic equations, systems of elliptic equations, and higher-order elliptic equations as well as isotropic and anisotropic diffusion problems are considered. The second chapter addresses stabilization to a stationary solution for nonlinear evolution problems. Here again, the authors progress from parabolic equations to anisotropic parabolic equations, to systems of equations of combined type and higher order parabolic equations. The third chapter addresses space and time localization of weak solutions for nonlinear degenerate parabolic equations as well as systems of these equations. Application of energy methods to various fluid mechanics problems is provided in the fourth chapter. The chapter begins with a brief review of theoretical fluid mechanics, followed by a series of applications ranging from free boundary gas dynamics problems such as free jets and collisions of co-axial jets to filtration of immiscible fluids in porous media which finds application in recovery of oil from a reservoir and to flows of non-homogeneous non-Newtonian fluids. The authors also treat a class of problems which require the simultaneous study of surface channel flow as well as underground water flow. The final two topics studied involves the drift diffusion model for free electrons that forms quasi-linear degenerate systems of PDEs and finds application in semi-conductor theory and the so-called blow up regimes in the solution of coupled PDEs for heat conduction and electrostatic potential resulting from the model of heat diffusion problems in a conductor in the presence of Joule heating. The book often follows mathematical exposition of statement of lemmas and theorems, followed by proofs and corollaries as well as relevant remarks. Each chapter terminates with a bibliographic review and a set of open problems that are essentially proposals for a program of research in topics not fully investigated in the literature or by the authors. There are some illustrations, all of which are of high quality. There are no illustrative computational results. There is an appendix which contains some basic definitions of function spaces and a brief review of elementary inequalities and imbedding theorems. There is a copious number of 300+ references which support the text and a detailed index. The monograph is the 48th volume in a series entitled Progress in Nonlinear Differential Equations and Their Applications, edited by H Brezis of the Mathematics Department of both Rutgers University in New Jersey, and the Universite Pierre and Marie Curie in Paris. The series is described as lying at the interface of pure and applied mathematics, and indeed this book falls into such a category. Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics can serve as a reference on the subject of energy methods when they are treated as part of mathematics post-graduate courses on partial differential equations; It is recommended for acquisition by university libraries as a quality addition to their mathematics collections.

Higher‐order boundary element methods for transient diffusion problems. Part I: Bounded flux formulation

AbstractDespite the significant number of publications on boundary element methods (BEM) for time‐dependent problems of heat diffusion, there still remain issues that need to be addressed, most importantly accuracy of the numerical modelling. Although very precise for steady‐state problems, the common boundary element methods applied to transient problems do not yield highly accurate numerical solutions. This paper investigates the reasons that prohibit achievement of a high level of accuracy for transient heat diffusion problems with continuous temperature and bounded heat flux solutions. In order to greatly enhance the commonly used boundary element formulations, we propose higher‐order time interpolation functions, including quadratic and quartic approximations. We show that the use of higher‐order time functions greatly reduces the numerical error concentrated in the corner regions, and results in very good uniformity of the flux and temperature distributions along the boundaries for problems where uniform distributions are expected.In order to highlight the importance of proper resolution both in time and space for the transient problems, we consider one‐ and two‐dimensional formulations in this paper. High‐order boundary elements using quartic shape functions, as well as high‐order bi‐quartic volume cells, are used to attain mesh‐independent numerical solutions. We consider four transient heat diffusion problems that possess exact solutions to investigate the convergence rate and accuracy of the higher‐order boundary element formulations. A very high level of accuracy is possible for both one‐ and two‐dimensional formulations. Additionally, we show that the accuracy of a commercially available finite‐element code is far less than that of the boundary element methods for a given spatial and temporal discretization. Copyright © 2002 John Wiley & Sons, Ltd.

Novel Transmission Open Photoacoustic Cell Configuration for Thermal Diffusivity Measurements in Liquids

In this paper the photoacoustic technique in the thermal-wave transmission configuration is applied to thermal diffusivity measurements in liquids. The one-dimensional heat diffusion problem involving three layers, and assuming surface absorption only, is solved for this goal. Linear relations among the photoacoustic amplitude (on a semi-log scale) and phase, as functions of the liquid sample thickness, are shown in each case. An analytical procedure involving linear fits to the experimental data is developed to produce two independent values for thermal diffusivity. The thermal diffusivity of three homogeneous liquids (distilled water, ethylene-glycol, and olive oil) was measured, and excellent agreement was obtained between results from both the amplitude and phase, as well as with thermal-diffusivity values reported in the literature.

Numerical oscillation in seepage analysis of unsaturated soils

The finite element method provides a popular means of analyzing groundwater flow in an unsaturated soil. In such problems, oscillatory results are often observed in the finite element solution. Such a phenomenon is observed, for example, when a typical finite element program such as Seep/w is used to model water infiltration into unsaturated soils. Numerical oscillations are often found near the wetting front where the hydraulic gradient is the steepest. These oscillations do not always reduce with decreasing or increasing time-step size alone; rather, an appropriate ratio between time-step size and element size is required. As the pore-water pressures predicted from a transient seepage analysis are used as input groundwater conditions for other types of analysis such as slope stability, contaminant transport, and capillary barrier, these oscillations may have important practical ramifications. Since seepage analysis is common in engineering practice, it is important that appropriate criteria are identified to minimize, if not to remove, the oscillations. In this paper, numerical examples are provided to demonstrate that a simple set of criteria, developed in heat diffusion problems with constant properties to control oscillation, is also applicable to one- and two-dimensional unsaturated seepage analyses, for a range of material nonlinearities that are frequently encountered in unsaturated soils.Key words: unsaturated soil, soil-water characteristic curve, seepage analysis, finite element method, numerical oscillation.

Numerical oscillation in seepage analysis of unsaturated soils

The finite element method provides a popular means of analyzing groundwater flow in an unsaturated soil. In such problems, oscillatory results are often observed in the finite element solution. Such a phenomenon is observed, for example, when a typical finite element program such as Seep/w is used to model water infiltration into unsaturated soils. Numerical oscillations are often found near the wetting front where the hydraulic gradient is the steepest. These oscillations do not always reduce with decreasing or increasing time-step size alone; rather, an appropriate ratio between time-step size and element size is required. As the pore-water pressures predicted from a transient seepage analysis are used as input groundwater conditions for other types of analysis such as slope stability, contaminant transport, and capillary barrier, these oscillations may have important practical ramifications. Since seepage analysis is common in engineering practice, it is important that appropriate criteria are identified to minimize, if not to remove, the oscillations. In this paper, numerical examples are provided to demonstrate that a simple set of criteria, developed in heat diffusion problems with constant properties to control oscillation, is also applicable to one- and two-dimensional unsaturated seepage analyses, for a range of material nonlinearities that are frequently encountered in unsaturated soils.Key words: unsaturated soil, soil-water characteristic curve, seepage analysis, finite element method, numerical oscillation.

EXTENSION OF THE METHOD OF EIGENFUNCTIONS TO THE BOUNDARY-VALUE PROBLEMS OF MECHANICAL DIFFUSION FOR MULTILAYER BODIES WITH INTERLAYERS

A method of eigenfunctions is developed for nonclassical axisymmetrical problems of heat and substance diffusion with generalized contact conditions in multilayer bodies. Based on a scalar product of a nonclassical form, the solutions of the above problems are constructed in the form of expansions in terms of eigenvalues. The solution of the problem of finding stresses caused by diffusion-type processes in multilayer bodies with interlayers is found.

Some Convergence Results for a Class of Nonlinear Phase-Field Evolution Equations

Two heat diffusion problems in the framework of the parabolic phase-field model are presented. The first problem is related to a single isotropic fluid and the other describes the heat transmission between two different substances in contact. Some known existence and uniqueness results are briefly recalled. Then, an asymptotic analysis of both situations is carried out as the kinetic equation collapses to a temperature-phase relation of Stefan type, in the first case in the whole material, and in the second in only one of the substances. In both cases, a convergence result for the solutions is proved. The second problem shows some more mathematical difficulties that are due to the presence of nontrivial terms on the common boundary. In order to control the latter, some tools are used from the Γ-convergence theory for convex functionals.

Complete and satisfactory solutions of luikov equations of heat and moisture transport in a spherical capillary — Porous body

Available results based on the exact solutions of the Luikov system of coupled heat and moisture diffusion problems addressed to the boundary conditions of the third kind for the bodies of simple geometry, have been reconsidered and revised because the transcendental equations have countable number of complex conjugate roots under certain conditions, which were ignored in the original computations. A methodology, quite different from Lobo et al., is presented for evaluating a set of countable number of pairs of complex conjugate roots. The influence of complex eigenvalues on the temperature and moisture distributions in a spherical capillary porous body is demonstrated and benchmark results are obtained for reference purpose.

Abstract Approach to Evolution Equations of Phase-Field Type and Applications

A general system of abstract nonlinear parabolic equations deriving from phase-field models of heat transfer in anisotropic fluids with phase transitions is studied through a backward finite differences approximation scheme and existence and uniqueness theorems are proved. Then, applications are presented to two physical situations: first, a standard heat diffusion problem is briefly discussed mainly as a justification of the choices of the abstract setting. Second, a more interesting transmission problem between two different adjoining fluids is studied under very general compatibility hypotheses and the abstract results are adapted to provide an existence and uniqueness theorem also in this case. Moreover, through convex analysis techniques, a mathematical study is performed of the subdifferential-type relations linking the phase fields and the temperatures to the internal energies of the fluids.