Non-Fourier Heat Conduction Problems and the Use of Exponential Basis Functions

A solution method using exponential basis functions (EBFs) is proposed for transient one-/two-dimensional non-Fourier heat conduction problems having particular application in bio-heat fields. A summation of EBFs satisfying the governing differential equation is considered in time and space. The presented method uses a noniterative algorithm for the solution of direct/inverse problems. It is demonstrated that the use of extra EBFs in the form of enrichment functions significantly improves the results when some jumps are seen in the input data. Four numerical examples, including bio-heat conduction problems, are provided to investigate the accuracy and performance of the method presented.     

Investigation for the dual phase lag behavior of bio-heat transfer

The success of hyperthermia treatment depends on the precise prediction and control of temperature distribution in the tissue. It was absolutely a necessity for hyperthermia treatment planning to understand the heat transport occurring in biological tissue. The tissue is highly non-homogenous, and non-Fourier thermal behavior in biological tissue has been experimentally observed. The dual phase lag model of heat conduction has been used to interpret the non-Fourier thermal behavior. This work attempts to be an extension study of Antaki [12] and explore whether the DPL thermal behavior exists in tissue. The inverse non-Fourier bio-heat transfer problem in the bi-layer spherical geometry is analyzed. In order to further address whether the dual phase lag model of bio-heat transfer merits additional study, the comparisons of the history of temperature increase among the present calculated results, the calculated values from the classical bio-heat transfer equation, and the experimental data are made for various measurement locations.     

A simple direct estimation of temperature-dependent thermal conductivity with kirchhoff transformation

A direct method is proposed to estimate the temperature-dependent thermal conductivity without internal measurements. In the proposed method, the steady-state nonlinear heat conduction equation is transformed into the Laplace equation via the Kirchhoff transformation. The thermal conductivity is modeled as a linear combination of known functions with unknown coefficients, which are directly determined from the imposed heat flux and measured temperatures at the boundary. Several inverse heat conduction problems are successfully introduced to confirm the validity of the proposed method.     

Shape identification for inverse geometry heat conduction problems by FEM without iteration

The boundary geometry shape is identified by the finite element method (FEM) without iteration and mesh reconstruction for two-dimensional (2-D) and three-dimensional (3-D) inverse heat conduction problems. First, the direct heat conduction problem with the exact domain is solved by the FEM and the temperatures of measurement points are obtained. Then, by introducing a virtual boundary, a virtual domain is formed. By minimizing the difference between the temperatures of measurement points in the exact domain and those in the virtual domain, the temperatures of the points on the virtual boundary are calculated based on the least square error method and the Tikhonov regularization. Finally, the objective geometry shape can be estimated by the method of searching the isothermal curve or isothermal surface for 2-D or 3-D problems, respectively. In the process, no iterative calculation is needed. The proposed method has a tremendous advantage in reducing the computational time for the inverse geometry problems. Numerical examples are presented to test the validity of the proposed approach. Meanwhile, the influences of measurement noise, virtual boundary, measurement point number, and measurement point position on the boundary geometry prediction are also investigated in the examples. The solutions show that the method is accurate and efficient to identify the unknown boundary geometry configurations for 2-D and 3-D heat conduction problems.     

Estimation for the heating effect of magnetic nanoparticles in perfused tissues

This work uses the transient bio-heat equation to predict the temperature rise behavior occurring in biological tissues during magnetic nanoparticles hyperthermia. A numerical scheme is proposed to solve the bio-heat transfer problem in a bi-layered spherical tissue with blood perfusion and metabolism. The analytical solution evidences the accuracy of the numerical scheme and examines the results in the literature.     

Reconstruction of the Space- and Time-Dependent Blood Perfusion Coefficient in Bio-Heat Transfer

The identification of the space- and time-dependent perfusion coefficient in the one-dimensional transient bio-heat conduction equation is investigated. While boundary and initial conditions are prescribed, additional temperature measurements are considered inside the solution domain. The problem is approached both from a global and a local perspective. In the global approach a Crank–Nicolson-type scheme is combined with the Tikhonov regularization method. In the local approach, we compute both the time first-order and space second-order derivatives by means of first kind integral equations. A comparison between the numerical results obtained using the two methods shows that the local approach is more accurate and stable than the global one.     

Temperature-Constrained Topology Optimization of Transient Heat Conduction Problems

A regional temperature measure model is constructed to obtain a small number of temperature constraints for local transient temperature control. The temperature sensitivity is derived using the adjoint variable method. The multiple temperature criteria and three-phase topology optimization are further investigated for transient heat conduction design. The material layout design of transient heat conduction is replaced by a static optimization problem, which is subsequently solved by the method of moving asymptotes. Finally, several numerical examples are provided to demonstrate the feasibility and validity of the proposed topology optimization for transient heat conduction problems.     

Resolution of Unknown Heat Source Inverse Heat Conduction Problems Using Particle Swarm Optimization

The main objective of this article is to solve inverse heat conduction problems with the particle swarm optimization method. An enhanced particle swarm optimization (EPSO) algorithm is proposed to overcome the shortcoming of earlier convergence of standard PSO algorithms. The EPSO is used to estimate the unknown time-dependent heat source in complex regions. Numerical experiments indicate the validity and stability of the EPSO method.     

Numerical analysis of an equivalent heat transfer coefficient in a porous model for simulating a biological tissue in a hyperthermia therapy

An equivalent heat transfer coefficient between tissue and blood in a porous model is investigated, which is applied to the thermal analysis of a biological tissue in a hyperthermia therapy. This paper applies a finite difference method to solving the tissue temperature distribution using Pennes’ bio-heat transfer equation and a two-equation porous model, respectively, and then employs a conjugate gradient method to estimate the equivalent heat transfer coefficient in the two-equation porous model with a known perfusion rate in Pennes’ bio-heat transfer equation. The results indicate that the equivalent heat transfer coefficient is not a strong function of the perfusion rate, blood velocity and heating conditions, but is inversely related to the blood vessel diameter.     

Boundary Element Method（BEM） for Solving Normal or Inverse Bio—heat Transfer Problem of Biological Bodies with Complex Shape

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Inverse Hyperbolic Heat Conduction in Fins with Arbitrary Profiles

This study aims to estimate unknown base temperature distribution in different non-Fourier fins. The Cattaneo–Vernotte (CV) heat model is used to predict the heat conduction behavior in these fins. This inverse problem is solved by the function-estimation version of the Adjoint conjugate gradient method (ACGM) based on boundary temperature measurements. The ACGM includes direct, sensitivity, and adjoint problems. For each of these problems, a one-dimensional general formulation of the non-Fourier model for longitudinal fins with arbitrary profile is driven and solved by an implicit finite difference method. In this study, three different profiles are considered: triangular, convex parabolic, and concave parabolic. For each of them, two different base temperature distributions are estimated using an inverse method. Moreover, the effects of sensor positions at the fin tip and a specific place in-between are considered on the base temperature estimation. A close agreement between the exact values and the estimated results is found, confirming the validity and accuracy of the proposed method. The results show that the ACGM is an accurate and stable method to determine the thermal boundary conditions in different non-Fourier fin problems.     

Virtual boundary element method in conjunction with conjugate gradient algorithm for three-dimensional inverse heat conduction problems

In this article, the virtual boundary element method (VBEM) in conjunction with conjugate gradient algorithm (CGA) is employed to treat three-dimensional inverse problems of steady-state heat conduction. On the one hand, the VBEM may face numerical instability if a virtual boundary is improperly selected. The numerical accuracy is very sensitive to the choice of the virtual boundary. The condition number of the system matrix is high for the larger distance between the physical boundary and the fictitious boundary. On the contrary, it is difficult to remove the source singularity. On the other hand, the VBEM will encounter ill-conditioned problem when this method is used to analyze inverse problems. This study combines the VBEM and the CGA to model three-dimensional heat conduction inverse problem. The introduction of the CGA effectively overcomes the above shortcomings, and makes the location of the virtual boundary more free. Furthermore, the CGA, as a regularization method, successfully solves the ill-conditioned equation of three-dimensional heat conduction inverse problem. Numerical examples demonstrate the validity and accuracy of the proposed method.     

COMPUTATION OF TURBULENT NATURAL CONVECTION IN A RECTANGULAR CAVITY WITH THE k–ϵ– –f MODEL

A hybrid numerical method involving the Laplace transform technique and finite-difference method in conjunction with the least-squares method and actual experimental temperature data inside the test material is proposed to estimate the unknown surface conditions of inverse heat conduction problems with the temperature-dependent thermal conductivity and heat capacity. The nonlinear terms in the differential equations are linearized using the Taylor series approximation. In this study, the functional form of the surface conditions is unknown a priori and is assumed to be a function of time before performing the inverse calculation. In addition, the whole time domain is divided into several analysis subtime intervals and then the unknown estimates on each subtime interval can be predicted. In order to show the accuracy and validity of the present inverse scheme, a comparison among the present estimates, direct solution, and actual experimental temperature data is made. The effects of the measurement errors, initial guesses, and measurement location on the estimated results are also investigated. The results show that good estimation of the surface conditions can be obtained from the present inverse scheme in conjunction with knowledge of temperature recordings inside the test material.     

Estimation of the periodic thermal conditions on the non-Fourier fin problem

A sequential method is proposed to estimate the periodic boundary conditions on the non-Fourier fin problem. An inverse solution is deduced from a finite difference method, the concept of the future time and a modified Newton-Raphson method. Two examples are used to demonstrate the features of the proposed method. The close agreement between the exact values and the estimated results is made to confirm the validity and accuracy of the proposed method. The results show that the proposed method is an accurate and stable method to determine the periodic boundary conditions in the non-Fourier fin problems.     

Nonlinear inverse heat conduction problem of surface temperature estimation by calibration integral equation method

A calibration integral equation method is proposed for estimating the surface temperature in the context of a nonlinear inverse heat conduction problem. The temperature-dependent thermophysical properties and probe positioning are implicitly accounted in the integral equation formulation through calibration tests. A first kind Chebyshev expansion is applied to represent the temperature-dependent property transform function. The undetermined expansion coefficients associated with the Chebyshev expansion are then estimated through two calibration tests. Regularization of the ill-posed problem is achieved by the future-time method. The optimal regularization parameter is estimated using a phase plane and cross-correlation phase plane analyses. Numerical simulation for stainless steel yields highly favorable surface temperature prediction.     

Inverse estimation of temperature boundary conditions with irregular shape of gas tank

To broaden the application of inverse estimation, the purpose of this study is to estimate the unknown temperature boundary condition of the complex or irregular shape, like the high pressure gas tank. An inverse algorithm based on the sequential method and the concept of future time combined with the finite-element method is proposed to solve the two dimensional irregular shape heat conduction problems. Special features about the proposed method are that the stiffness matrix of the irregular shape can be solved from the finite-element method and used by the inverse algorithm. The estimated results are quite accurate with the consideration of future time in the different measured errors, the various sensor’s number and the sensor location. These results show that the proposed method is an accurate, sturdy, and efficient method for solving several realistic applications.     

SOLUTION OF STEADY PERIODIC HEAT CONDUCTION BY THE FINITE-ELEMENT METHOD

This paper describes the use of the finite-element technique to solve problems of steady periodic heal conduction. The concept of a complex variable can be used to reduce the governing unsteady heat conduction equation to two noncoupled Poisson equations. The validity of the present method is confirmed with a one-dimensional problem for which an analytic solution exists. Numerical solutions for a three-dimensional problem are presented to illustrate the capability of the method.     

Inverse problem of unsteady conjugated forced convection in parallel plate channels

The heat transfer phenomena of the unsteady laminar forced convection in parallel plate channels with wall conduction effects are still not very well understood. An inverse algorithm based on the conjugate gradient method is proposed to estimate the boundary conditions of these problems, and the minimization of object function is used to reduce the estimated error. The estimation of applied heat flux is found to be highly dependent of temperature sensor location and uncertainty, plate thickness, and heating way. The results show that the predicted boundary conditions by the present inverse method are consistent with the initially specified ones.     

Estimation of two-sided boundary conditions for two-dimensional inverse heat conduction problems

A hybrid numerical algorithm of the Laplace transform technique and finite-difference method with a sequential-in-time concept and the least-squares scheme is proposed to predict the unknown surface temperature of two-sided boundary conditions for two-dimensional inverse heat conduction problems. In the present study, the functional form of the estimated surface temperatures is unknown a priori. The whole time domain is divided into several analysis sub-time intervals and then the unknown surface temperatures in each analysis interval are estimated. To enhance the accuracy and efficiency of the present method, a good comparison between the present estimations and previous results is demonstrated. The results show that good estimations on the surface temperature can be obtained from the transient temperature recordings only at a few selected locations even for the case with measurement errors. It is worth mentioning that the unknown surface temperature can be accurately estimated even though the thermocouples are located far from the estimated surface. Owing to the application of the Laplace transform technique, the unknown surface temperature distribution can be estimated from a specific time.     

Heterogeneous Heat Conduction Problems by an Improved Element-Free Galerkin Method

In this article an improved element-free Galerkin method is proposed to solve heat conduction problems with heterogeneous media. Because the method almost possesses interpolation property, the implementation of essential boundary condition is as simple as that in the finite-element method. In order to validate the proposed method, several heat conduction problems with different degrees of heterogeneity are presented. In these test problems, we focus on the influence of nodal distribution to the proposed method for heat conduction problems with heterogeneous media. It is shown that, for different degrees of heterogeneity, regardless of matter whether the node is located on the interface, accurate solutions can be obtained by the proposed method for heterogeneous heat conduction problems without a source term.