Meshless analysis of three-dimensional steady-state heat conduction problems

Steady-state heat conduction problems arisen in connection with various physical and engineering problems where the functions satisfy a given partial differential equation and particular boundary conditions, have attracted much attention and research recently. These problems are independent of time and involve only space coordinates, as in Poisson＇s equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, it is difficult to find an analytical solution, the only choice left is an approximate numerical solution. This paper deals with the numerical solution of three-dimensional steady-state heat conduction problems using the meshless reproducing kernel particle method （RKPM）. A variational method is used to obtain the discrete equations. The essential boundary conditions are enforced by the penalty method. The effectiveness of RKPM for three-dimensional steady-state heat conduction problems is investigated by two numerical examples.

The Improved Element-Free Galerkin Method for Anisotropic Steady-State Heat Conduction Problems

In this paper,we considered the improved element-free Galerkin(IEFG)method for solving 2D anisotropic steadystate heat conduction problems.The improved moving least-squares(IMLS)approximation is used to establish the trial function,and the penalty method is applied to enforce the boundary conditions,thus the final discretized equations of the IEFG method for anisotropic steady-state heat conduction problems can be obtained by combining with the corresponding Galerkin weak form.The influences of node distribution,weight functions,scale parameters and penalty factors on the computational accuracy of the IEFG method are analyzed respectively,and these numerical solutions show that less computational resources are spent when using the IEFG method.

Meshless Least-Squares Method for Solving the Steady-State Heat Conduction Equation

The meshless weighted least-squares (MWLS) method is a pure meshless method that com- bines the moving least-squares approximation scheme and least-square discretization. Previous studies of the MWLS method for elastostatics and wave propagation problems have shown that the MWLS method possesses several advantages, such as high accuracy, high convergence rate, good stability, and high com- putational efficiency. In this paper, the MWLS method is extended to heat conduction problems. The MWLS computational parameters are chosen based on a thorough numerical study of 1-dimensional problems. Several 2-dimensional examples show that the MWLS method is much faster than the element free Galerkin method (EFGM), while the accuracy of the MWLS method is close to, or even better than the EFGM. These numerical results demonstrate that the MWLS method has good potential for numerical analyses of heat transfer problems.

Isogeometric Boundary Element Method for Two-Dimensional Steady-State Non-Homogeneous Heat Conduction Problem

The isogeometric boundary element technique(IGABEM)is presented in this study for steady-state inhomogeneous heat conduction analysis.The physical unknowns in the boundary integral formulations of the governing equations are discretized using non-uniform rational B-spline(NURBS)basis functions,which are utilized to build the geometry of the structures.To speed up the assessment of NURBS basis functions,the Bezier extraction´approach is used.To solve the extra domain integrals,we use a radial integration approach.The numerical examples show the potential of IGABEM for dimension reduction and smooth integration of CAD and numerical analysis.

The Method of Fundamental Solutions for Steady-State Heat Conduction in Nonlinear Materials

The steady-state heat conduction in heat conductors with temperature dependent thermal conductivity and mixed boundary conditions involving radiation is investigated using the method of fundamental solutions.Various computational issues related to the method are addressed and numerical results are presented and discussed for problems in two and three dimensions.

Finite Difference-Peridynamic Differential Operator for Solving Transient Heat Conduction Problems

Transient heat conduction problems widely exist in engineering.In previous work on the peridynamic differential operator(PDDO)method for solving such problems,both time and spatial derivatives were discretized using the PDDO method,resulting in increased complexity and programming difficulty.In this work,the forward difference formula,the backward difference formula,and the centered difference formula are used to discretize the time derivative,while the PDDO method is used to discretize the spatial derivative.Three new schemes for solving transient heat conduction equations have been developed,namely,the forward-in-time and PDDO in space(FT-PDDO)scheme,the backward-in-time and PDDO in space(BT-PDDO)scheme,and the central-in-time and PDDO in space(CT-PDDO)scheme.The stability and convergence of these schemes are analyzed using the Fourier method and Taylor’s theorem.Results show that the FT-PDDO scheme is conditionally stable,whereas the BT-PDDO and CT-PDDO schemes are unconditionally stable.The stability conditions for the FT-PDDO scheme are less stringent than those of the explicit finite element method and explicit finite difference method.The convergence rate in space for these three methods is two.These constructed schemes are applied to solve one-dimensional and two-dimensional transient heat conduction problems.The accuracy and validity of the schemes are verified by comparison with analytical solutions.

Galerkin-based quasi-smooth manifold element(QSME)method for anisotropic heat conduction problems in composites with complex geometry

The accurate and efficient analysis of anisotropic heat conduction problems in complex composites is crucial for structural design and performance evaluation. Traditional numerical methods, such as the finite element method(FEM), often face a trade-off between calculation accuracy and efficiency. In this paper, we propose a quasi-smooth manifold element(QSME) method to address this challenge, and provide the accurate and efficient analysis of two-dimensional(2D) anisotropic heat conduction problems in composites with complex geometry. The QSME approach achieves high calculation precision by a high-order local approximation that ensures the first-order derivative continuity.The results demonstrate that the QSME method is robust and stable, offering both high accuracy and efficiency in the heat conduction analysis. With the same degrees of freedom(DOFs), the QSME method can achieve at least an order of magnitude higher calculation accuracy than the traditional FEM. Additionally, under the same level of calculation error, the QSME method requires 10 times fewer DOFs than the traditional FEM. The versatility of the proposed QSME method extends beyond anisotropic heat conduction problems in complex composites. The proposed QSME method can also be applied to other problems, including fluid flows, mechanical analyses, and other multi-field coupled problems, providing accurate and efficient numerical simulations.

Variational Approach to 2D and 3D Heat Conduction Modeling

The paper proposes an approximate solution to the classical (parabolic) multidimensional 2D and 3D heat conduction equation for a 5 × 5 cm aluminium plate and a 5 × 5 × 5 cm aluminum cube. An approximate solution of the generalized (hyperbolic) 2D and 3D equation for the considered plate and cube is also proposed. Approximate solutions were obtained by applying calculus of variations and Euler-Lagrange equations. In order to verify the correctness of the proposed approximate solutions, they were compared with the exact solutions of parabolic and hyperbolic equations. The paper also presents the research on the influence of time parameters τ as well as the relaxation times τ ∗ to the variation of the profile of the temperature field for the considered aluminum plate and cube.

Variational Approach to Heat Conduction Modeling

It is known that Fourier’s heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. This is unacceptable on physical grounds in spite of the fact that Fourier’s law agrees well with experiment. However, discrepancies are likely to occur when extremely short distances or extremely short time intervals are considered, as they must in some modern problems of aero-thermodynamics. Cattaneo and independently Vernotte proved that such process can be described by Heaviside’s telegraph equation. This paper shows that this fact can be derived using calculus of variations, by application of the Euler-Lagrange equation. So, we proved that the equation of heat conduction with finite velocity propagation of the thermal disturbance can be obtained as a solution to one variational problem.

Sterilization Effect of Cooking Process for Guilin Rice Noodles Based on Heat Conduction Model

Guilin rice noodles, a unique cuisine from Guilin, Guangxi, is renowned both domestically and internationally as one of the top ten “Guilin Classics”. Utilizing a heat conduction model, this study explores the effectiveness of the cooking process in sterilizing Guilin rice noodles before consumption. The model assumes that a large pot is filled with boiling water which is maintained at a constant high temperature heat resource through continuous gentle heating. And the room temperature is set as the initial temperature for the preheating process and the final temperature for the cooling process. The objective is to assess whether the cooking process achieves satisfactory sterilization results. The temperature distribution function of rice noodle with time is analytically obtained using the separation of variables method in the three-dimensional cylindrical coordinate system. Meanwhile, the thermal diffusion coefficient of Guilin rice noodles is obtained in terms of Riedel’ theory. By analyzing the elimination characteristics of Pseudomonas cocovenenans subsp. farinofermentans, this study obtains the optimal time required for effective sterilization at the core of Guilin rice noodles. The results show that the potential Pseudomonas cocovenenans subsp. farinofermentans will be completely eliminated through continuously preheating more than 31 seconds during the cooking process before consumption. This study provides a valuable reference of food safety standards in the cooking process of Guilin rice noodles, particularly in ensuring the complete inactivation of potentially harmful strains such as Pseudomonas cocovenenans subsp. farinofermentans.

Voronoi Based Discrete Least Squares Meshless Method for Heat Conduction Simulation in Highly Irregular Geometries

A new technique is used in Discrete Least Square Meshfree（DLSM） method to remove the common existing deficiencies of meshfree methods in handling of the problems containing cracks or concave boundaries. An enhanced Discrete Least Squares Meshless method named as VDLSM（Voronoi based Discrete Least Squares Meshless） is developed in order to solve the steady-state heat conduction problem in irregular solid domains including concave boundaries or cracks. Existing meshless methods cannot estimate precisely the required unknowns in the vicinity of the above mentioned boundaries. Conducted researches are limited to domains with regular convex boundaries. To this end, the advantages of the Voronoi tessellation algorithm are implemented. The support domains of the sampling points are determined using a Voronoi tessellation algorithm. For the weight functions, a cubic spline polynomial is used based on a normalized distance variable which can provide a high degree of smoothness near those mentioned above discontinuities. Finally, Moving Least Squares（MLS） shape functions are constructed using a varitional method. This straight-forward scheme can properly estimate the unknowns（in this particular study, the temperatures at the nodal points） near and on the crack faces, crack tip or concave boundaries without need to extra backward corrective procedures, i.e. the iterative calculations for modifying the shape functions of the nodes located near or on these types of the complex boundaries. The accuracy and efficiency of the presented method are investigated by analyzing four particular examples. Obtained results from VDLSM are compared with the available analytical results or with the results of the well-known Finite Elements Method（FEM） when an analytical solution is not available. By comparisons, it is revealed that the proposed technique gives high accuracy for the solution of the steady-state heat conduction problems within cracked domains or domains with concave boundaries and at the same time possesses a high convergence rate which its accuracy is not sensitive to the arrangement of the nodal points. The novelty of this paper is the use of Voronoi concept in determining the weight functions used in the formulation of the MLS type shape functions.

Decay rates and time lags of heat conduction in building construction under field conditions

The field measurements of decay rates and time lags of heat conduction in a building construction taken in Nanjing during the summer of 2001 are presented.The decay rates and time lags are calculated according to the frequency responses of the heat absorbed by the room＇s internal surfaces,inside surface temperature,indoor air temperature and outdoor synthetic temperature.The measured results match very well with the theoretical results of the zeroth and the first order values of the decay rates and time lags of heat conduction in the building construction,but the difference between the measured values and the theoretical values for the second order is too great to be accepted.It is therefore difficult to accurately test the second order value.However,it is still advisable to complete the analysis using the zeroth-and the first-orders values of the decay rates and time lags of heat conduction in building construction under field conditions,because in these cases the decay rates of heat conduction reach twenty which meets the requirements of engineering plans.

Solution of an Inverse Problem of Heat Conduction of 45 Steel with Martensite Phase Transformation in High Pressure during Gas Quenching

In order to simulate thermal strains, thermal stresses, residual stresses and microstructure of the steel during gas quenching by means of the numerical method, it is necessary to obtain an accurate boundary condition of temperature field. The surface heat transfer coefficient is a key parameter. The explicit finite difference method, nonlinear estimation method and the experimental relation between temperature and time during gas quenching have been used to solve the inverse problem of heat conduction. The relationship between surface temperature and surface heat transfer coefficient of a cylinder has been given. The nonlinear surface heat transfer coefficients include the coupled effects between martensitic phase transformation and temperature.

HYBRID FEM WITH FUNDAMENTAL SOLUTIONS AS TRIAL FUNCTIONS FOR HEAT CONDUCTION SIMULATION

A new type of hybrid finite element formulation with fundamental solutions as internal interpolation functions, named as HFS-FEM, is presented in this paper and used for solving two dimensional heat conduction problems in single and multi-layer materials. In the proposed approach, a new variational functional is firstly constructed for the proposed HFS-FE model and the related existence of extremum is presented. Then, the assumed internal potential field constructed by the linear combination of fundamental solutions at points outside the elemental domain under consideration is used as the internal interpolation function, which analytically satisfies the governing equation within each element. As a result, the domain integrals in the variational functional formulation can be converted into the boundary integrals which can significantly simplify the calculation of the element stiffness matrix. The independent frame field is also introduced to guarantee the inter-element continuity and the stationary condition of the new variational functional is used to obtain the final stiffness equations. The proposed method inherits the advantages of the hybrid Trefftz finite element method （HT-FEM） over the conventional finite element method （FEM） and boundary element method （BEM）, and avoids the difficulty in selecting appropriate terms of T-complete functions used in HT-FEM, as the fundamental solutions contain usually one term only, rather than a series containing infinitely many terms. Further, the fundamental solutions of a problem are, in general, easier to derive than the T-complete functions of that problem. Finally, several examples are presented to assess the performance of the proposed method, and the obtained numerical results show good numerical accuracy and remarkable insensitivity to mesh distortion.

Fast precise integration method for hyperbolic heat conduction problems

A fast precise integration method is developed for the time integral of the hyperbolic heat conduction problem. The wave nature of heat transfer is used to analyze the structure of the matrix exponential, leading to the fact that the matrix exponential is sparse. The presented method employs the sparsity of the matrix exponential to improve the original precise integration method. The merits are that the proposed method is suitable for large hyperbolic heat equations and inherits the accuracy of the original version and the good computational efficiency, which are verified by two numerical examples.

Analytical solutions of transient heat conduction in multilayered slabs and application to thermal analysis of landfills

The study of transient heat conduction in multilayered slabs is widely used in various engineering fields. In this paper, the transient heat conduction in multilayered slabs with general boundary conditions and arbitrary heat generations is analysed. The boundary conditions are general and include various combinations of Dirichlet, Neumann or Robin boundary conditions at either surface. Moreover, arbitrary heat generations in the slabs are taken into account. The solutions are derived by basic methods, including the superposition method, separation variable method and orthogonal expansion method. The simplified double-layered analytical solution is validated by a numerical method and applied to predicting the temporal and spatial distribution of the temperature inside a landfill. It indicates the ability of the proposed analytical solutions for solving the wide range of applied transient heat conduction problems.

Fuzzy finite difference method for heat conduction analysis with uncertain parameters

A new numerical technique named as fuzzy finite difference method is proposed to solve the heat conduction problems with fuzzy uncertainties in both the phys- ical parameters and initial/boundary conditions. In virtue of the level-cut method, the difference discrete equations with fuzzy parameters are equivalently transformed into groups of interval equations. New stability analysis theory suited to fuzzy difference schemes is developed. Based on the parameter perturbation method, the interval ranges of the uncertain temperature field can be approximately predicted. Subsequently, fuzzy solutions to the original difference equations are obtained by the fuzzy resolution theorem. Two numerical examples are given to demonstrate the feasibility and efficiency of the presented method for solving both steady-state and transient heat conduction problems.

A new complex variable meshless method for transient heat conduction problems

In this paper, based on the improved complex variable moving least-square （ICVMLS） approximation, a new complex variable meshless method （CVMM） for two-dimensional （2D） transient heat conduction problems is presented. The variational method is employed to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. As the transient heat conduction problems are related to time, the Crank-Nicolson difference scheme for two-point boundary value problems is selected for the time discretization. Then the corresponding formulae of the CVMM for 2D heat conduction problems are obtained. In order to demonstrate the applicability of the proposed method, numerical examples are given to show the high convergence rate, good accuracy, and high efficiency of the CVMM presented in this paper.

Thermoelastic waves in helical strands with Maxwell–Cattaneo heat conduction

Harmonic thermoelastic waves in helical strands with Maxwell–Cattaneo heat conduction areinvestigated analytically and numerically. The corresponding dispersion relation is a sixth-orderalgebraic equation, governed by six non-dimensional parameters: two thermoelastic couplingconstants, one chirality parameter, the ratio between extensional and torsional moduli, the Fouriernumber, and the dimensionless thermal relaxation. The behavior of the solutions is discussedfrom two perspectives with an asymptotic-numerical approach: (1) the effect of thermal relaxationon the elastic wave celerities, and (2) the effect of thermoelastic coupling on the thermal wavecelerities. With small wavenumbers, the adiabatic solution for Fourier helical strands is recovered.However, with large wavenumbers, the solutions behave differently depending on the thermalrelaxation and chirality. Due to thermoelastic coupling, the thermal wave celerity deviates from theclassical result of the speed of second sound.

Increment-Dimensional Scaled Boundary Finite Element Method for Solving Transient Heat Conduction Problem

An increment-dimensional scaled boundary finite element method (ID-SBFEM) is developed to solve the transient temperature field.To improve the accuracy of SBFEM,the effect of high frequency factor on dynamic stiffness is considered,and the first-order continued fraction technique is used.After the derivation,the SBFE equations are obtained,and the dimensions of thermal conduction,the thermal capacity matrix and the vector of the right side term in the equations are doubled.An example is presented to illustrate the feasibility and good accuracy of the proposed method.