Virtual Boundary Meshless with Trefftz Method for the Steady-State Heat Conduction Crack Problem

This article presents a virtual boundary meshless with Trefftz method for calculation of the two-dimensional steady-state heat conduction crack problem, which is based on the idea of multidomain combination. The virtual source function is constructed by the radial basis functions approximation in the subdomain not containing the crack. The temperature or heat flux is calculated by crack special solution of Trefftz function in the crack subdomain. Thus, the proposed method has the advantages of both the boundary-type meshless method and the Trefftz method. No nodes or elements are distributed on the crack, because the boundary condition of the crack is automatically satisfied by employing the crack special solution of the Trefftz function. The heat stress intensity factor is simply and directly calculated.     

Modified Collocation Trefftz Method for the Geometry Boundary Identification Problem of Heat Conduction

In this article, a meshless numerical algorithm is proposed for the boundary identification problem of heat conduction, one kind of inverse problem. In the geometry boundary identification problem, the Cauchy data is given for part of the boundary. The Neumann boundary condition is given for the other portion of the boundary, whose spatial position is unknown. In order to stably solve the inverse problem, the modified collocation Trefftz method, a promising boundary-type meshless method, is adopted for discretizing this problem. Since the spatial position for part of the boundary is unknown, the numerical discretization results in a system of nonlinear algebraic equations (NAEs). Then, the exponentially convergent scalar homotopy algorithm (ECSHA) is used to efficiently obtain the convergent solution of the system of NAEs. The ECSHA is insensitive to the initial guess of the evolutionary process. In addition, the efficiency of the computation is greatly improved, since calculation of the inverse of the Jacobian matrix can be avoided. Four numerical examples are provided to validate the proposed meshless scheme. In addition, some factors that might influence the performance of the proposed scheme are examined through a series of numerical experiments. The stability of the proposed scheme can be proven by adding some noise to the boundary conditions.     

A Precise Integration Boundary-Element Method for Solving Transient Heat Conduction Problems with Variable Thermal Conductivity

In this article, a combined approach of the radial integration boundary element method (RIBEM) and the precise integration method is presented for solving transient heat conduction problems with variable thermal conductivity. First, the system of ordinary differential equations on the boundary integral equation can be obtained by the RIBEM. Then, the precise integration method is adopted to solve the system of ordinary differential equations. Finally, three numerical examples are presented to demonstrate the performance of the present method. The results show that the present approach can obtain satisfactory performance even for very large time-step size.     

Numerical Simulation of Heat Conduction Problems by a New Fast Multipole Hybrid Boundary-Node Method

The fast multipole method (FMM) is an effective technique to reduce the computational cost in solving large-scale problems. In this article, a new fast multipole hybrid boundary-node method (FM-HBNM) is presented to solve three-dimensional heat conduction problems. In the new FM-HBNM, a diagonal form for translation operators is used and the computational cost of the multipole to local (M2L) translation is further reduced. Formulations for the new FM-HBNM are derived. The computational costs for the original and new FM-HBNM are estimated. The numerical results show that a speed-up about 2–3 times can be achieved by the new FM-HBNM.     

A Precise Time-Domain Expanding Boundary-Element Method for Solving Three-Dimensional Transient Heat Conduction Problems with Variable Thermal Conductivity

A combined approach of the radial integration boundary-element method (RIBEM) and the precise algorithm in the time domain is presented for solving three-dimensional transient heat conduction problems with variable thermal conductivity. First, by expanding physical quantities at discrete time intervals, the recursive formulation of the governing equation is derived. Then, the recursive equation is solved by the RIBEM, and a self-adaptive check technique is carried out to estimate how many expansion terms are needed in a time step. Finally, three numerical examples show that the present approach can obtain very stable and accurate results for different time-step size.     

Radiation Element Method Coupled with the Lattice Boltzmann Method Applied to the Analysis of Transient Conduction and Radiation Heat Transfer Problem with Heat Generation in a Participating Medium

This article deals with the implementation of the radiation element method (REM) with the lattice Boltzmann method (LBM) to solve a combined mode transient conduction-radiation problem. Radiative information computed using the REM is provided to the LBM solver. The planar conducting-radiating participating medium is contained between diffuse gray boundaries, and the system may contain a volumetric heat generation source. Temperature and heat flux distributions in the medium are studied for different values of parameters such as the extinction coefficient, the scattering albedo, the conduction-radiation parameter, the emissivity of the boundaries, and the heat generation rate. To check the accuracy of the results, the problem is also solved using the finite-volume method (FVM) in conjunction with the LBM. In this case, the data for radiation field are calculated using the FVM. The REM has been found to be compatible with the LBM, and in all the cases, results of the LBM-REM and the LBM-FVM have been found to provide an excellent comparison.     

A Global Heat Compliance Measure Based Topology Optimization for the Transient Heat Conduction Problem

For topology optimization with transient loads, heat compliance varies with transient heat analysis. The peak value of the transient heat compliance should be minimized. Thus, this article proposes a global heat compliance measure to handle this kind of topology optimization for the transient heat conduction problem. The optimization model is then constructed by the global heat compliance measure. The finite-element, equivalent static loads, and continuum shape based sensitivity analyses are derived using the adjoint variable method. Through case studies, the effectiveness of the proposed global heat compliance measure for the transient heat conduction topology optimization is validated.     

Lattice Boltzmann Method Applied to the Solution of Energy Equation of a Radiation and Non-Fourier Heat Conduction Problem

This article concerns the application of the lattice Boltzmann method (LBM) to solve the energy equation of a combined radiation and non-Fourier conduction heat transfer problem. The finite propagation speed of the thermal wave front is accounted by non-Fourier heat conduction equation. The governing energy equation is solved using the LBM. The finite-volume method (FVM) is used to compute the radiative information. The formulation is validated by taking test cases in 1-D planar absorbing, emitting, and scattering medium whose west boundary experiences a sudden rise in temperature, or, with adiabatic boundaries, the medium is subjected to a sudden localized energy source. Results are analyzed for the various values of parameters like the extinction coefficient, the scattering albedo, the conduction-radiation parameter, etc., on temperature distributions in the medium. Radiation has been found to help in facilitating faster distribution of energy in the medium. Unlike Fourier conduction, wave fronts have been found to reflect from the boundaries. The LBM-FVM combination has been found to provide accurate results.     

A New High-Precision Boundary-Type Meshless Method for Calculation of Multidomain Steady-State Heat Conduction Problems

This article presents the idea for calculating 2-D steady-state heat conduction problems with multidomain combination by employing the virtual boundary meshless least-square method. Being different from the conventional virtual boundary-element method (VBEM), this method incorporates the point interpolation method (PIM) with the compactly supported radial basis function (CSRBF) to approximately construct the virtual source function of the VBEM. Thus, the proposed method has the advantages of both the boundary-type meshless method and the virtual boundary element method. Since the configuration of the virtual boundary requires a certain preparation, the integration along the virtual boundary can be carried out over the smooth simple curve that can be structured beforehand (for 2-D problems) to reduce the complexity and difficulty of calculus without loss of accuracy, while the “vertex question” existing in the BEM can be avoided. Numerical examples show that the proposed method is more precise than several other numerical methods while selecting fewer degrees of freedom. In addition, its numerical stability is also verified by computing several cases.     

Two-Dimensional Heat Conduction Problem in a Functionally Graded Plate with a Slanting Boundary to the Functional Gradation

This article presents analytical results for temperature in a functionally graded material plate (FGMP) with a slanting boundary to the functional gradation subjected to a partial heating. The heat conductivity is expressed in terms of a exponential function of the position. The general solution of the heat conduction equation for FGMP with a slanting boundary to the functional gradation is derived by use of the variable separation method and the analytical solution, which satisfies the boundary condition is obtained. Numerical calculations are carried out for ZrO₂/Ti-6Al-4V and ZrO₂/stainless (SUS304) functionally graded plates, when the ceramic surface is partially heated. Temperature and heat flux are graphically displayed for these two cases.     

On the Solution Of Heterogeneous Heat Conduction Problems by a Diffuse Approximation Meshless Method

A diffuse approximation-based collocation method is used to solve a two-dimensional heterogeneous heat conduction problem. It is shown that one has to take into account the neighbors of each neighbor of the calculation node in order to achieve sufficient accuracy in the case of strong heterogeneity.     

The Method of Particular Solutions for Solving Inverse Problems of a Nonhomogeneous Convection-Diffusion Equation with Variable Coefficients

A new version of the method of particular solutions (MPS) has been proposed for solving inverse problems for nonhomogeneous convection-diffusion equations with variable coefficients (IPCD). Coupled with the time discretization and MPS, the proposed method is a truly meshless method which requires neither domain or boundary discretization. Even though the final temperature is almost undetectable or is disturbed by significant noise, the proposed method can still recover the initial temperature very well. The effectiveness of the proposed inverse scheme using radial basis functions is demonstrated by several examples in 2-D and 3-D.     

Solution of an Inverse Heat Conduction Problem in a Bi-Layered Spherical Tissue

This work attempts to estimate the phase lag times of a tissue based on the dual-phase-lag model from the experimental data. The inverse dual-phase-lag bioheat transfer problem in the bilayered spherical tissue is studied. The difference between two layers in the thermophysical parameters, geometry effects, and measurement errors of the input data make it hard to be solved. To solve the present problem, a hybrid scheme based on the Laplace transform, change of variables, and the least-squares scheme is proposed. In order to evidence the validity and accuracy of the estimated results, the comparison of the history of temperature increase between the calculated results and the experimental data is made for various measurement locations. The effect of measurement location on the estimated results is also investigated.     

An Iterative Method to Recover the Heat Conductivity Function of a Nonlinear Heat Conduction Equation

The left-boundary data of temperature and heat flux are used to estimate an unknown heat conductivity function in a nonlinear heat conduction equation. A Lie-group adaptive method (LGAM) is developed to derive a Lie-group equation defined at two different times, which can be used to recover a spatial-dependence heat conductivity function through a few iterations. Also, the one-dimensional Calderón inverse problem is addressed by applying the present methodology to recover a steady-state heat conductivity coefficient in terms of space variable. The convergence speed and accuracy of the present LGAM are examined by linear and nonlinear numerical examples under noisy input data.     

Model Reduction for Heat Conduction with Radiative Boundary Conditions using the Modal Identification Method

A model reduction for a conductive material with radiative boundary condition is carried out using the modal identification method. For this type of system, we show that the classical application of the modal identification method may lead to nonacceptable reduced models as a result of some low sensitivities and parameter correlations. Hence, a new reduced model formulation is stated to improve the model reduction process. Some numerical test cases illustrate the study. It is found that the reduced models fit very well with the original system, reducing the order up to 99.5%.     

Finite-Element Discretized Symplectic Method for Steady-State Heat Conduction with Singularities in Composite Structures

A new-finite element discretized symplectic method for solving the steady-state heat conduction problem with singularities in composite structures is presented. The model with a singularity is divided into two regions, near and far fields, and meshed using conventional finite elements. In the near field, the temperature and heat flux densities are expanded in exact symplectic eigensolutions. After a matrix transformation, the unknowns in the near field are transformed to coefficients of the symplectic series, while those in the far field are as usual. The exact local solutions for temperature and heat flux densities are obtained simultaneously without any post-processing.     

Homotopy Perturbation Method for Solving the Two-Phase Inverse Stefan Problem

In this article, the possibility of the application of the homotopy perturbation method for solving the two-phase inverse Stefan problem is presented. This problem consists in the calculation of temperature distribution in the domain, as well as in the reconstruction of the functions describing the temperature and the heat flux on the boundary when the position of the moving interface is known. The validity of the approach is verified by comparing the results obtained with the exact solution.     

根据润滑理论求解轴承特性的新方法

介绍轴承润滑的流体力学理论、无限长轴承的特性方程式及其数值解法。