Heat Source Problem Research Articles

The necessary optimal condition of inverse heat source problem

In this paper, we study the inverse problem of finding the source for heat conduction from the integral observation and the final time one. Using least-square method, this problem is reformulated into the optimal one of minimizing the objective functional. The Fréchet differentiability and the derivative of the objective functional are derived via an adjoint problem. Some numerical examples for finding the necessary optimality conditions are presented.

New thermal packaging with a boiling-driven heat spreader for thermal management of the power electronics

• A boiling-driven heat spreader for the power conversion system. • Thermal management performance was compared among 3 heat spreader designs. • Universal support design is a key design factor for multiple heat source problems. • A numerical methodology was proposed for the boiling-driven heat spreader. A new thermal management device with a two-phase flat heat spreader was designed, fabricated, and characterized. The boiling-driven heat spreader (280 mm × 65 mm × 4 mm) is embedded at the thermal packaging air-cooling heat sink for the power electronics. A sintered microporous structure was coated on the surface in the heat source's direction to promote a boiling heat transfer inside the heat spreader. The thermal management devices are tested with the potential applications consisting of the six heat sources placed vertically. As a result, boiling-driven heat spreaders have proven to be an equivalent thermal management method in applications with multiple heat sources and high heat flux targets. In particular, the heat spreader equipped with a dot-type supporter can make a temperature difference between junctions within 7.6℃ at 1500 W in case of air cooling. Compared to the result (Δ T = 28℃ @ 1500 W) of the solid copper plate of the same size, this thermal packaging demonstrated improved thermal management performance. In addition, we propose a numerical analysis for a boiling-driven heat spreader. The numerical analysis results can predict multiple heat sources' temperature distribution more accurately rather than assume a two-phase heat spreader as a solid with high thermal conductivity. This study provides valuable insight into the thermal management and thermal packaging design of power electronic devices. It represents a breakthrough in resolving the reliability problem of thermal packaging technology and temperature non-uniformity.

Analytical Model of Heating an Isotropic Half-Space by a Moving Laser Source with a Gaussian Distribution

This study presents the solution of the transient spatial problem of the impact of a moving source of heat flux induced by laser radiation on the surface of a half-space using the superposition principle and the method of transient functions. The solution is based on the Green’s function method, according to which the influence function of a surface-concentrated heat source is found at the first stage. The influence function has axial symmetry and the problem of finding the influence function is axisymmetric. To find the Green’s function, Laplace and Fourier integral transforms are used. The novelty of the obtained analytical solution is that the heat transfer at the free surface of the half-space is taken into account. The Green’s function that was obtained is used to construct an analytical solution to the moving heat-source problem in the integral form. The kernel of the advising integral operator is the constructed Green’s function. The Gaussian distribution is used to calculate integrals on spatial variables analytically. Gaussian law models the distribution of heat flux in the laser beam. As a result, the corresponding integrals on the spatial variables can be calculated analytically. A convenient formula that allows one to study the non-stationary temperature distribution when the heat source moves along arbitrary trajectories is obtained. A numerical, analytical algorithm has been developed and implemented that allows one to determine temperature distribution both on the surface and on the depth of a half-space. For verification purposes, the results were compared with the solution obtained using FEM.

Recovering Heat Source from Fourth-Order Inverse Problems by Weighted Gradient Collocation

The weighted gradient reproducing kernel collocation method is introduced to recover the heat source described by Poisson’s equation. As it is commonly known that there is no unique solution to the inverse heat source problem, the weak solution based on a priori assumptions is considered herein. In view of the fourth-order partial differential equation (PDE) in the mathematical model, the high-order gradient reproducing kernel approximation is introduced to efficiently untangle the problem without calculating the high-order derivatives of reproducing kernel shape functions. The weights of the weighted collocation method for high-order inverse analysis are first determined. In the benchmark analysis, the unclear illustration in the literature is clarified, and the correct interpretation of numerical results is given particularly. Two mathematical formulations with four examples are provided to demonstrate the viability of the method, including the extreme cases of the limited accessible boundary.

Unicity of solution for a semi-infinite inverse heat source problem

<abstract><p>A semi-infinite inverse source problem in heat conduction equations is considered, where the source term is assumed to be compactly supported in the region. After introducing a suitable artificial boundary, the semi-infinite problem is transformed into a bounded one and the corresponding exact expression of the boundary condition is derived. Then we rigorously prove the uniqueness of the solution of original problem, together with the stability of the corresponding optimal control solution.</p></abstract>

An adaptive wavelet collocation method for the optimal heat source problem

PurposeThis paper aims to propose a new adaptive numerical method to find more accurate numerical solution for the heat source optimal control problem (OCP).Design/methodology/approachThe main aim of this paper is to present an adaptive collocation approach based on the interpolating wavelets to solve an OCP for finding optimal heat source, in a two-dimensional domain. This problem arises when the domain is heated by microwaves or by electromagnetic induction.FindingsThis paper shows that combination of interpolating wavelet basis and finite difference method makes an accurate structure to design adaptive algorithm for such problems which usually have non-smooth solution.Originality/valueThe proposed numerical technique is flexible for different OCP governed by a partial differential equation with box constraint over the control or the state function.

Temperature response for active thermography using laser scanning heating and method of images

When scanning a specimen using a line heat source at a constant velocity, a temperature change occurs partly due to specular reflection at the defect interface. Assuming that the reflection of a transient thermal response is similar to that of geometrical optics, we performed waveform analysis using an imaging method. Image points are calculated based on differential geometry; this can also be performed for a curved surface using a general equation. A combination of the steepest descent analysis of a moving heat source problem and a convolution technique successfully yielded waveforms comparable to those of experimental temperature responses. We designed and constructed an active thermographic imaging system in which a linearly focused continuous wave laser beam was scanned perpendicular to the beam as it covered the entire surface of specimens with simulated internal defects. The real-time response was recorded as a temperature waveform at each image pixel. Waveforms were calculated for specimens without or with buried cylindrical defects parallel to their surfaces and compared to the experimental data. The theory well-explains the signal generation mechanism, and excellent agreement was obtained in waveforms. Some discrepancy between theory and experiment indicates more complicated problems in heat and mass transfer.

Development of avocado shape heat source model for finite element based heat transfer analysis of high-velocity arc welding process

The mathematical representation of the welding heat source is a critical task as the competence of the model decides the prediction accuracy of the thermal profile of the welded joint. The geometric shape of the moving heat source in fusion welding is a strong function of weld velocity. The elliptical shape conforms to the non-symmetric volumetric energy distribution by a double-ellipsoidal heat source model at low speed. However, the geometric conformity of the elongated weld pool may not be elliptical at relatively high welding speed. In the present work, a new volumetric heat source model, the so-called avocado shape, is proposed, a more realistic approximation of the applied volumetric heat flux and flexible enough to accommodate a wide variation of welding conditions. The proposed model is a modified version of ellipsoidal shape with non-symmetrical energy distribution and unequal axis length in the front and rear parts along the weld velocity vector. Thus, it is different from the double-ellipsoidal model in terms of user-defined parameters and from the egg-configuration model in terms of unequal axis lengths. The avocado shape heat source model is self-consistent since it estimates all the parameters from weld width and depth of penetration. The heat source follows the Gaussian distribution of volumetric energy flux for a moving heat source problem. A finite element based heat transfer analysis is performed using the avocado shape heat source model for gas tungsten arc welding (GTAW) and submerged arc welding (SAW) processes. Compared with the experimentally measured weld dimensions, time-temperature history and microstructure evaluate the prediction accuracy of the proposed heat source model.

The Ritz–Galerkin procedure for an inverse space-dependent heat source problem

In this paper, an inverse problem of recovering the unknown space-dependent source term in a parabolic equation under a final overdetermination condition is considered. To keep matters simple, in our analysis the main problem has been considered in the one-dimensional case, however the proposed method can be applied for higher dimensional cases. The approximate solution of the inverse problem is implemented by the Ritz–Galerkin method. The shifted Legendre polynomials basis together with the Galerkin approach are employed to reduce the main problem to the solution of linear algebraic equations. To overcome the difficulties arising from solving the resultant ill-conditioned linear system, a type of regularization technique is utilized to obtain a stable solution. The convergence analysis of the suggested method using Gronwall’s inequality is studied. Finally, some numerical examples are provided to demonstrate the efficiency and applicability of the proposed algorithm in the presence of noise in input measured data.

Inverse Heat Source Problem and Experimental Design for Determining Iron Loss Distribution

Iron loss determination in the magnetic core of an electrical machine, such as a motor or a transformer, is formulated as an inverse heat source problem. The sensor positions inside the object are ...

Using asymptotic expansions technique for solving the point heat source problem in the fractional theory of thermoelasticity with the Caputo Fabrizio derivative

In this work, the Caputo Fabrizio definition of fractional derivative is utilized to explore a problem in thermoelasticity with fractional theory. The problem regards an infinite elastic space under the impact of an uninterrupted heat point source. The method of asymptotic expansions is used to find the inverse Laplace transform valid for short times. A summarized discussion about the behavior of wave’s propagation is introduced.

A homogenization function method for inverse heat source problems in 3D functionally graded materials

A simple and effective method is proposed for solving inverse heat source problems in functionally graded materials based on the homogenization function. Making use of given conditions, a homogenization function for the boundary value problem is conceived and a family of homogenization functions is further derived. Then, the superposition of homogenization functions method is developed and used for determining the heat source of the inverse problems. In this new methodology, the inverse heat source problems are directly solved by calculating a linear matrix system. Importantly, this scheme does not involve mesh generation, numerical integration, iteration, regularization and fundamental solutions, and it is easy to program and implement on the existing software. Four numerical examples defined on the cuboid domains are presented to demonstrate the accuracy and efficiency of the presented tool.

Stability results for identifying an unknown source term of a heat equation in the Banach space L1(Rn)

In this paper, we prove a stability estimate of H¨older type and propose a regularization method with error estimates of H¨older type for an inverse heat source problem in the Banach space L1(Rn).

A Radial Basis Function Collocation Method for Space-dependent Inverse Heat Problems

In this study, a radial basis function collocation method (RBFCM) is proposed for the numerical treatment of inverse space-wise dependent heat source problems. Multiquadric radial basis function is applied for spatial discretization whereas for temporal discretization Runge-Kutta method of order four is employed. Numerical experiments for one, two and three-dimensional cases are included to test the efficiency and accuracy of the suggested method. Both non-rectangular and rectangular geometries with uniform and non-uniform points are taken into consideration and the obtained results are compared with the exact as well as with the techniques presented in recent literature.

A local meshless method for the numerical solution of space‐dependent inverse heat problems

In this paper, a local radial basis function collocation method is proposed for the numerical solution of inverse space‐wise dependent heat source problems. Multiquadric radial basis function is used for spatial discretization. The method accuracy is tested in terms of absolute root mean square and relative root mean square error norms. Numerical tests on a noisy data are performed on both regular domain and irregular domain. To test the efficiency and accuracy of the proposed method, numerical experiments for one‐, two‐, and three‐dimensional cases are performed. Both regular and irregular geometries with uniform and nonuniform points are taken into consideration, and the numerical results are also compared with the existing methods reported in literature.

An efficient and accurate method for transient heat conduction in a periodic structure with moving heat sources

PurposeThis paper aims to propose an efficient and accurate method to analyse the transient heat conduction in a periodic structure with moving heat sources.Design/methodology/approachThe moving heat source is modelled as a localised Gaussian distribution in space. Based on the spatial distribution, the physical feature of transient heat conduction and the periodic property of structure, a special feature of temperature responses caused by the moving heat source is illustrated. Then, combined with the superposition principle of linear system, within a small time-step, computation of results corresponding to the whole structure excited by the Gaussian heat source is transformed into that of some small-scale structures. Lastly, the precise integration method (PIM) is used to solve the temperature responses of each small-scale structure efficiently and accurately.FindingsWithin a reasonable time-step, the heat source applied on a unit cell can only cause the temperature responses of a limited number of adjacent unit cells. According to the above feature and the periodic property of a structure, the contributions caused by the moving heat source for the most of time-steps are repeatable, and the temperature responses of the entire periodic structure can be obtained by some small-scale structures.Originality/valueA novel numerical method is proposed for analysing moving heat source problems, and the numerical examples demonstrate that the proposed method is much more efficient than the traditional methods, even for larger-scale problems and multiple moving heat source problems.

Heat Conduction Simulation of 2D Moving Heat Source Problems Using a Moving Mesh Method

This paper focuses on efficiently numerical investigation of two-dimensional heat conduction problems of material subjected to multiple moving Gaussian point heat sources. All heat sources are imposed on the inside of material and assumed to move along some specified straight lines or curves with time-dependent velocities. A simple but efficient moving mesh method, which continuously adjusts the two-dimensional mesh dimension by dimension upon the one-dimensional moving mesh partial differential equation with an appropriate monitor function of the temperature field, has been developed. The physical model problem is then solved on this adaptive moving mesh. Numerical experiments are presented to exhibit the capability of the proposed moving mesh algorithm to efficiently and accurately simulate the moving heat source problems. The transient heat conduction phenomena due to various parameters of the moving heat sources, including the number of heat sources and the types of motion, are well simulated and investigated.

Numerical Investigation for the Arrangement of Perforated Tile under a Non-uniform Heat Source Condition in a Raised-floor Data Center

The research object of this paper was a typical raised-floor data center where the heat load of each rack was non-uniform. The non-uniform heat source caused that the temperature field in the room was also non-uniform and part of the rack outlet temperature was relatively high. In view of this problem, a numerical simulation was conducted to investigate the effect of the arrangement of perforated tile on the air distribution and the temperature field. The results indicate that a better arrangement of perforated tile could optimize the air distribution and make the temperature field in the room more uniform. The maximum rack outlet temperature was reduced by about 2.1°C under the proper condition of the tiles opening ratio, which means that the energy efficiency of the cooling system was accordingly improved. The adjustment of the tiles opening ratio could be a solution to the non-uniform heat source problem in a raised-floor data center.

Solving the higher-dimensional nonlinear inverse heat source problems by the superposition of homogenization functions method

The paper solves the higher-dimensional inverse heat source problems of nonlinear convection-diffusion-reaction equations in 2D rectangles and 3D cuboids, of which the final time data and the Neumann boundary data on one-side are over-specified. Firstly, we derive a family of single-parameter space-time homogenization functions. Secondly, the temperature is obtained through the superposition of the homogenization functions and solving a linear system to satisfy the over-specified Neumann boundary condition. Thirdly, the unknown heat source is recovered by the back substitution of the numerical solution of temperature into the nonlinear governing equation. Numerical tests reveal that the novel method is very accurate to find the solution and to recover the unknown space-time dependent heat source function in the whole space–time domain, whose required extra data are very saving.

Uniqueness for an inverse problem for a nonlinear parabolic system with an integral term by one-point Dirichlet data

We consider an inverse problem arising in laser-induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues is destroyed by coagulation. For the dosage planning quantitatively reliable numerical simulation are indispensable. To this end the identification of the thermal growth kinetics of the coagulated zone is of crucial importance. Mathematically, this problem is a nonlinear and nonlocal parabolic inverse heat source problem. We show in this paper that the temperature dependent thermal growth parameter can be identified uniquely from a one-point measurement.