Time-dependent Heat Source Research Articles

Simultaneous reconstruction of the time-dependent Robin coefficient and heat flux in heat conduction problems

This paper aims to solve an inverse heat conduction problem in two-dimensional space under transient regime, which consists of the estimation of multiple time-dependent heat sources placed at the boundaries. Robin boundary condition (third type boundary condition) is considered at the working domain boundary. The simultaneous identification problem is formulated as a constrained minimization problem using the output least squares method with Tikhonov regularization. The properties of the continuous and discrete optimization problem are studied. Differentiability results and the adjoint problems are established. The numerical estimation is investigated using a modified conjugate gradient method. Furthermore, to verify the performance of the proposed algorithm, obtained results are compared with results obtained from the well-known finite-element software COMSOL Multiphysics under the same conditions. The numerical results show that the proposed algorithm is accurate, robust and capable of simultaneously representing the time effects on reconstructing the time-dependent Robin coefficient and heat flux.

Dynamic problem in 3D thermoelastic half-space with rotation in context of G-N type II and type III

In this paper, comparison between G-N model of type II (without energy dissipation) and G-N model of type III (with energy dissipation) has been shown in a three dimensional thermoelastic half space with rotation subjected to time dependent heat source on the traction free boundary. Eigenvalue methodology has been adopted to solve the equations resulting from the application of the Normal mode analysis to the non-dimensional coupled equations. Variation of the numerically computed values of thermal stresses and temperature with and without rotation has been illustrated graphically.

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.

Meshless Local B-Spline Collocation Method for Two-Dimensional Heat Conduction Problems With Nonhomogenous and Time-Dependent Heat Sources

This paper presents a new approach of meshless local B-spline based finite difference (FD) method for transient 2D heat conduction problems with nonhomogenous and time-dependent heat sources. In this method, any governing equations are discretized by B-spline approximation which is implemented as a generalized FD technique using local B-spline collocation scheme. The key aspect of the method is that any derivative is stated as neighboring nodal values based on B-spline interpolants. The set of neighboring nodes is allowed to be randomly distributed. This allows enhanced flexibility to be obtained in the simulation. The method is truly meshless as no mesh connectivity is required for field variable approximation or integration. Galerkin implicit scheme is employed for time integration. Several transient 2D heat conduction problems with nonuniform heat sources in arbitrary complex geometries are examined to show the efficacy of the method. Comparison of the simulation results with solutions from other numerical methods in the literature is given. Good agreement with reference numerical methods is obtained. The method is shown to be simple and accurate for the time-dependent problems.

Temperature Dependence of the Elastic Modulus in Three-Dimensional Generalized Thermoelasticity with Dual-Phase-Lag Effects

A three-dimensional problem for a homogeneous isotropic thermoelastic half-space solids with temperature-dependent mechanical properties subject to a time-dependent heat sources on the boundary of the half-space which is traction free is considered in the context of the generalized thermoelasticity with dual-phase-lag effects. The normal mode analysis and eigenvalue approach techniques are used to solve the resulting non-dimensional coupled field equations. Numerical results for the temperature, thermal stresses and displacement distributions are represented graphically and discussed. A comparison is made with the result obtained in the absence of the temperature dependent elastic modulus. Various problems of generalized thermoelasticity and conventional coupled dynamical thermoelasticity are deduced as special cases of our problem.

Recovering a general space-time-dependent heat source by the coupled boundary integral equation method

ABSTRACTAs to recover a time-dependent heat source under an extra temperature measured at an interior point, we can reformulate it to be a three-point boundary value problem. We can develop a coupled boundary integral equation method, wherein by selecting two sets of adjoint test eigenfunctions in two sub domains and using polynomials as the trial functions of unknown heat source, the time-dependent heat source is recovered very well and quickly. Four numerical examples, including a discontinuous one, demonstrate the efficiency for the ill-posed inverse heat source problem in a large time duration and under a large noise up to 10–30%. Then, selecting three sets of adjoint test eigenfunctions in three domains: problem domain and two sub domains, and using the Pascal polynomials as trial functions, the unknown space-time-dependent heat source is recovered very fast and accurately from the solution of three coupled boundary integral equations.

On an influence of thermal stresses and magnetic field in thermoelastic half-space without energy dissipation

ABSTRACTIn this article, a two-dimensional problem for a homogeneous, isotropic, and thermoelastic half-space subjected to magnetic field and time-dependent heat source is investigated. The boundary conditions at the surface have been considered in the context of Green-Naghdi’s second model (GN-II) of thermoelasticity. The normal mode analysis and eigenvalue approach techniques are used to solve the nondimensional coupled equations. The effect of magnetic field, frequency, wave number, and time is analyzed theoretically and computed numerically. Comparison was made with the results obtained in the presence and absence of the magnetic field. Numerical results for the displacement components, mean value of normal stresses, dilatation, and temperature have been represented graphically to show the physical meaning of the external parameters. The results indicate that the effect of magnetic field, frequency, wave number, and time is very pronounced and effective on the phenomena.

Convective conditions on magnetohydrodynamic flow over stretched cylinder with time and space dependent heat source or sink

Convective conditions on magnetohydrodynamic flow over stretched cylinder with time and space dependent heat source or sink

Eigenvalue approach to two dimensional coupled magneto-thermoelasticity in a rotating isotropic medium

This paper is devoted to two-dimensional problem of thermoelasticity dealing with thermoelastic wave propagation in a rotating medium with magnetic field effects and a time-dependent heat source effect due to thermomechanical source. The normal mode analysis and eigenvalue approach techniques are applied to solve the problem. The expressions of displacement, mean value of normal stress, dilatation and temperature are obtained in the domain. Numerical simulated results are depicted graphically to show the effect of magnetic field and rotation on resulting quantities. The results indicate that the effect of magnetic field, rotation, frequency, wave number and time are very pronounced.

Plane wave propagation in a 3D anisotropic half-space under Green-Naghdi theory II

In this paper, the theory of coupled thermoelasticity in three dimension is employed for triclinic half-space, subjected to time dependent heat source on the boundary of the space which is traction free and is considered in the context of Green-Naghdi model of type II (thermoelasticity without energy dissipation) of generalized thermoelasticity. Normal mode analysis is used to the non-dimensional coupled equations. Finally, the resulting equations are written in the form of a vector-matrix differential equation which is then solved by eigenvalue approach. Numerical results for the temperature, thermal stresses, and displacements are presented graphically and analyzed. Mathematical results shown in thermoelastic curves were supplemented by tectonic movements of elastic lithospheric plates.

Simultaneous determination of time-dependent coefficients and heat source

ABSTRACTThis article presents a numerical solution to the inverse problems of simultaneous determination of the time-dependent coefficients and the source term in the parabolic heat equation subject to overspecified conditions of integral type. The ill-posed problems are numerically discretized using the finite-difference method. The resulting system of nonlinear equations is solved numerically using the MATLAB toolbox routine lsqnonlin applied to minimizing the nonlinear Tikhonov regularization functional subject to simple physical bounds on the variables. Numerical examples are presented to illustrate the accuracy and stability of the solution.

Recovery of a time-dependent heat source in one-dimensional thermoelasticity of type-III

In this contribution, the reconstruction of a solely time-dependent heat source is studied in an inverse problem arising in one-dimensional thermoelasticity. The missing heat source is recovered from a measurement in time of the average temperature inside the body. Using this observation, the inverse problem is reformulated in a direct setting. The existence and uniqueness of a weak solution are addressed, and a new numerical algorithm based on Rothe’s method is designed. Numerical experiments support the obtained results.

Analytical Solution for Three-Dimensional Hyperbolic Heat Conduction Equation with Time-Dependent and Distributed Heat Source

AbstractThis paper is devoted to the analytical solution of three-dimensional hyperbolic heat conduction equation in a finite solid medium with rectangular cross-section under time dependent and non-uniform internal heat source. The closed form solution of both Fourier and non-Fourier profiles are introduced with Eigen function expansion method. The solution is applied for simple simulation of absorption of a continues laser in biological tissue. The results show the depth of laser absorption in tissue and considerable difference between the Fourier and Non-Fourier temperature profiles. In addition the solution can be applied as a verification branch for other numerical solutions.

Axi-symmetric generalized thermoelastic diffusion problem with two-temperature and initial stress under fractional order heat conduction

A mathematical model of fractional order two-temperature generalized thermoelasticity with diffusion and initial stress is proposed to analyze the transient wave phenomenon in an infinite thermoelastic half-space. The governing equations are derived in cylindrical coordinates for a two dimensional axi-symmetric problem. The analytical solution is procured by employing the Laplace and Hankel transforms for time and space variables respectively. The solutions are investigated in detail for a time dependent heat source. By using numerical inversion method of integral transforms, we obtain the solutions for displacement, stress, temperature and diffusion fields in physical domain. Computations are carried out for copper material and displayed graphically. The effect of fractional order parameter, two-temperature parameter, diffusion, initial stress and time on the different thermoelastic and diffusion fields is analyzed on the basis of analytical and numerical results. Some special cases have also been deduced from the present investigation.

Identification of separable sources for advection-diffusion equations with variable diffusion coefficient from boundary measured data

This paper considers the inverse problems of identifying either the unknown space-dependent heat source or the unknown time-dependent heat source of the variable coefficient advection–diffusion equation with separable sources of the form from supplementary time-dependent temperature measurement at the right boundary of the domain. We proved that solutions of both inverse source problems can be identified uniquely under some regularity assumptions. Two non-iterative inversion algorithms were developed and numerically implemented for the identification of the unknown space-wise and time-wise dependent sources. The robustness and limitations of the two algorithms are investigated through numerical examples related to the reconstruction of continuous and discontinuous sources.

A Gaussian RBFs method with regularization for the numerical solution of inverse heat conduction problems

In this paper, a recursion numerical technique is considered to solve the inverse heat conduction problems, with an unknown time-dependent heat source and the Neumann boundary conditions. The numerical solutions of the heat diffusion equations are constructed using the Gaussian radial basis functions. The details of algorithms in the one-dimensional and two-dimensional cases, involving the global or partial initial conditions, are proposed, respectively. The Tikhonov regularization method, with the generalized cross-validation criterion, is used to obtain more stable numerical results, since the linear systems are badly ill-conditioned. Moreover, we propose some results of the condition number estimates to a class of positive define matrices constructed by the Gaussian radial basis functions. Some numerical experiments are given to show that the presented schemes are favourably accurate and effective.

A mollification regularization method for identifying the time-dependent heat source problem

We study in this paper the problem of determining time-dependent source functions in a parabolic equation with data given at some fixed locations in the domain. To solve this ill-posed inverse problem, we develop a mollification regularization method with a Gaussian kernel. We derive an a priori error estimate between the exact solution and its regularized approximation. Moreover, we propose an a posteriori parameter choice strategy for the selection of the regularization strength and derive an error estimate associated with the strategy. Numerical results are presented to illustrate the accuracy and efficiency of our method.

A new approximate method for an inverse time-dependent heat source problem using fundamental solutions and RBFs

This paper presents a meshless numerical scheme to solve the inverse heat source time dependent problem. Fundamental solutions of heat equations and radial basis functions (RBFs) are used to obtain a numerical solution. Since the coefficient matrix may be ill-conditioned, the Tikhonov regularization (TR) method is employed to solve the resulting system of linear equations. Therefore, the generalized cross-validation (GCV) criterion is applied to choose a regularization parameter. The accuracy and efficiency of the proposed method is illustrated by some numerical examples.

The multiple-scale polynomial Trefftz method for solving inverse heat conduction problems

The polynomial Trefftz method consists of the polynomial type solutions as bases, providing a cheap boundary-type meshless method to solve the heat conduction equation, since the bases automatically satisfy the governing equation. In order to stably solve the backward heat conduction problem (BHCP), and the inverse heat source problem (IHSP) together with the boundary condition recovery problem by a polynomial Trefftz method, which are both known to be highly ill-posed, we introduce a multiple-scale post-conditioner in the resultant linear system to reduce the condition number. Then the conjugate gradient method (CGM) is used to solve the post-conditioned linear system to determine the unknown expansion coefficients. In the multiple-scale polynomial Trefftz method (MSPTM) the scales are determined a priori by the collocation points on space–time boundary, which can retrieve the missing initial data, the unknown time-dependent heat source as well as the boundary condition rather well. Several numerical examples of the inverse heat conduction problems demonstrate that the MSPTM is effective and accurate, even for those of severely ill-posed inverse problems under very large noises.

Non-Fourier heat conduction analysis of infinite 2D orthotropic FG hollow cylinders subjected to time-dependent heat source

The non-Fourier heat conduction analysis of infinite 2D functionally graded (FG) hollow cylinders subjected to time-dependent heat source is presented. In this study, a new augmented state space method considering laminate approximation theory is introduced. All material properties are assumed to vary continuously within the cylinder along the specified directions following an arbitrary law. We considered acomplicated heat transfer engineering problem that so far has no analytical and/or numerical solution yet. This problem is solved in the Laplace domain, and to convert the obtained results into the time domain, a numerical Laplace transform inversion with consideration of Gibb's phenomenon is employed. This present method is proposed so general that can solve the heat conduction problems time-dependent heat source and general boundary conditions such as heat flux, convection and combinations of arbitrary temperatures. Eventually, our method is superior in correctness and exactness and also has a sufficient computational cost by comparing the obtained results with some available problems in the literatures.