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The aim of this paper is to provide a simple yet accurate and efficient geometric method for thermal homogenization of impregnated and non-impregnated coil winding technologies based on the concept of thermal resistance.

For regular windings, the periodic microscopic cell in the winding space is identified. Also, for irregular windings, the average microscopic cell of the winding is determined. An approximation is used to calculate the thermal resistance of the winding cell. Based on this approximation, the winding insulation is considered as a circular ring around the wire. Mathematical equations are obtained to calculate the equivalent thermal resistance of the cell. The equivalent thermal conductivity of the winding is calculated using equivalent thermal resistance of the cell. Winding thermal homogenization is completed by determining the equivalent thermal properties of the cell.

The thermal pattern of different windings is simulated and compared with the results of different homogenization methods. The results show that the proposed method is applicable for a wide range of windings in terms of winding scheme, packing factor and winding insulation. Also, the results show that the proposed method is more accurate than other winding homogenization methods in calculating the equivalent thermal conductivity of the winding.

In this paper, the change of electrical resistance of the winding with temperature and thermal contact between the sub-components are ignored. Also, liquid insulators, such as oils, and rectangular wires were not investigated. Research in these topics is considered as future work.

Unlike other homogenization methods, the proposed method can be applied to non-impregnated and irregular windings. Also, compared to other homogenization methods, the proposed method has a simpler formulation that makes it easier to program and implement. All of these indicate the efficiency of the proposed method in the thermal analysis of the winding.

In this paper, a homogenization‐based multi‐scale method for predicting the effective thermal conductivity of porous materials with radiation is presented, which considers the effect of geometry and distribution of pores. Using homogenization method to solve the pure conductive problem of porous materials with periodic structure, the effective thermal conductivity without considering radiation is predicted, and a temperature field in a local domain of a unit cell is obtained. This temperature field is taken as the good approximation of the real temperature distribution, and the radiative thermal conductivity is obtained. The effect of the microstructure, the distribution and geometry of pores on heat transfer of porous materials is discussed. It is concluded that the dimension of the pores is an important influence factor on the thermal transfer property of porous materials if radiation is considered. Increasing the pore’s dimension enhances the contribution of radiation to the heat transfer property of porous materials. For porous materials with cylindrical and spherical pores, the radiative thermal conductivity is proportional to pore’s diameter.

Effectiveness of the homogenization method for various heat transfer problems of engineering composites is the main aim of the paper. This comparative study is done for layered, fiber and particle reinforced Representative Volume Elements (RVE) for composites made of widely used components. Mathematical model is based on the effective modules method introduced for periodic composites ‐ effective heat conductivity is calculated in the closed form for specific spatial distribution of the components, while effective volumetric heat capacity is obtained from a simple spatial averaging. Such a homogenization scheme makes possible to significantly simplify the numerical analysis of transient heat transfer phenomena in various types of composites. The comparison of temperature histories obtained for the real and homogenized composite models is carried out using the Finite Element Method system ANSYS. As is demonstrated for various boundary problems, a homogenization technique in terms of composites types collected in the paper give satisfactory agreement with the real structure modeling; further numerical studies on composite cells discretization should increase modeling efficiency and diminish the numerical errors.

The optimal material microstructures in pure material design are no longer efficient or optimal when accounting macroscopic structure performance with specific boundary conditions. Therefore, it is important to provide a novel multiscale topology optimization framework to tailor the topology of structure and the material to achieve specific applications. In comparison with porous materials, composites consisting of two or more phase materials are more attractive and advantageous from the perspective of engineering application. This paper aims to provide a novel concurrent topological design of structures and microscopic materials for thermal conductivity involving multi-material topology optimization (material distribution) at the lower scale.

In this work, the effective thermal conductivity properties of microscopic three or more phase materials are obtained via homogenization theory, which serves as a bridge of the macrostructure and the periodic material microstructures. The optimization problem, including the topological design of macrostructures and inverse homogenization of microscopic materials, are solved by bi-directional evolutionary structure optimization method.

As a result, the presented framework shows high stability during the optimization process and requires little iterations for convergence. A number of interesting and valid macrostructures and material microstructures are obtained in terms of optimal thermal conductive path, which verify the effectiveness of the proposed mutliscale topology optimization method. Numerical examples adequately consider effects of initial guesses of the representative unit cell and of the volume constraints of adopted base materials at the microscopic scale on the final design. The resultant structures at both the scales with clear and distinctive boundary between different phases, making the manufacturing straightforward.

This paper presents a novel multiscale concurrent topology optimization method for structures and the underlying multi-phase materials for thermal conductivity. The authors have carried out the concurrent multi-phase topology optimization for both 2D and 3D cases, which makes this work distinguished from existing references. In addition, some interesting and efficient multi-phase material microstructures and macrostructures have been obtained in terms of optimal thermal conductive path.

The simulation of heat conduction inside a heterogeneous material with multiple spatial scales would require extremely fine and ill-conditioned meshes and, therefore, the success of such a numerical implementation would be very unlikely. This is the main reason why this paper aims to calculate an effective thermal conductivity for a multi-scale heterogeneous medium.

The methodology integrates the theory of reiterated homogenization with the finite element method, leading to a renewed calculation algorithm.

The effective thermal conductivity gain of the considered three-scale array relative to the two-scale array has been evaluated for several different values of the global volume fraction. For gains strictly above unity, the results indicate that there is an optimal local volume fraction for a maximum heat conduction gain.

The present approach is formally applicable within the asymptotic limits required by the theory of reiterated homogenization.

It is expected that the present analytical-numerical methodology will be a useful tool to aid interpretation of the gain in effective thermal conductivity experimentally observed with some classes of heterogeneous multi-scale media.

The novel aspect of this paper is the application of the integrated algorithm to calculate numerical bulk effective thermal conductivity values for multi-scale heterogeneous media.

This paper aims to present thermal and mechanical stresses in solid and hollow thick-walled cylinders and spheres made of functionally graded materials (FGMs) under the effect of heat generation.

Constant internal temperature and convective external conditions in hollow bodies along with internal heat generation with a combination of outer convective conditions in solid bodies are investigated individually. The effect of the heat convection coefficient on solid bodies is additionally discussed. The variation of the FGM properties in the radial direction is adapted to the Mori–Tanaka homogenization schemes, which produces irregular and two-point linear boundary value problems that are numerically solved by the pseudospectral Chebyshev method.

It has been shown that the selection of the mixtures of FGMs has to be made correctly to keep the thermal and mechanical loads acting on objects at low levels.

In this study, both solid and hollow functionally graded cylinders and spheres for different boundary conditions that are as their engineering applications are examined with the proposed method. The results have demonstrated that the pseudospectral Chebyshev method has high accuracy, low calculation costs and ease of application and can be easily adapted to such engineering problems.

This paper aims to establish a multiscale topology optimization method for the optimal design of non-periodic, self-supporting cellular structures subjected to thermo-mechanical loads. The result is a hierarchically complex design that is thermally efficient, mechanically stable and suitable for additive manufacturing (AM).

The proposed method seeks to maximize thermo-mechanical performance at the macroscale in a conceptual design while obtaining maximum shear modulus for each unit cell at the mesoscale. Then, the macroscale performance is re-estimated, and the mesoscale design is updated until the macroscale performance is satisfied.

A two-dimensional Messerschmitt Bolkow Bolhm (MBB) beam withstanding thermo-mechanical load is presented to illustrate the proposed design method. Furthermore, the method is implemented to optimize a three-dimensional injection mold, which is successfully prototyped using 420 stainless steel infiltrated with bronze.

By developing a computationally efficient and manufacturing friendly inverse homogenization approach, the novel multiscale design could generate porous molds which can save up to 30 per cent material compared to their solid counterpart without decreasing thermo-mechanical performance.

This study is a useful tool for the designer in molding industries to reduce the cost of the injection mold and take full advantage of AM.

Intumescent coatings are nowadays a dominant passive system used to protect structural materials in case of fire. Due to their reactive swelling behaviour, intumescent coatings are particularly complex materials to be modelled and predicted, which can be extremely useful especially for performance-based fire safety designs. In addition, many parameters influence their performance, and this challenges the definition and quantification of their material properties. Several approaches and models of various complexities are proposed in the literature, and they are reviewed and analysed in a critical literature review.

Analytical, finite-difference and finite-element methods for modelling intumescent coatings are compared, followed by the definition and quantification of the main physical, thermal, and optical properties of intumescent coatings: swelled thickness, thermal conductivity and resistance, density, specific heat capacity, and emissivity/absorptivity.

The study highlights the scarce consideration of key influencing factors on the material properties, and the tendency to simplify the problem into effective thermo-physical properties, such as effective thermal conductivity. As a conclusion, the literature review underlines the lack of homogenisation of modelling approaches and material properties, as well as the need for a universal modelling method that can generally simulate the performance of intumescent coatings, combine the large amount of published experimental data, and reliably produce fire-safe performance-based designs.

Due to their limited applicability, high complexity and little comparability, the presented literature review does not focus on analysing and comparing different multi-component models, constituted of many model-specific input parameters. On the contrary, the presented literature review compares various approaches, models and thermo-physical properties which primarily focusses on solving the heat transfer problem through swelling intumescent systems.

The presented literature review analyses and discusses the various modelling approaches to describe and predict the behaviour of swelling intumescent coatings as fire protection for structural materials. Due to the vast variety of available commercial products and potential testing conditions, these data are rarely compared and combined to achieve an overall understanding on the response of intumescent coatings as fire protection measure. The study highlights the lack of information and homogenisation of various modelling approaches, and it underlines the research needs about several aspects related to the intumescent coating behaviour modelling, also providing some useful suggestions for future studies.

To prevent the coil from burning or getting damaged, it is necessary to estimate the duration of its operation as long as its temperature does not exceed the permissible limit. This paper aims to investigate the effect of switching on the thermal behavior of impregnated and nonimpregnated windings. Also, the safe operating time for each winding is determined.

The power loss of the winding is expressed as a function of the winding specifications. Using homogenization techniques, the equivalent thermal properties for the homogenized winding are calculated and used in a proposed thermal equivalent circuit for winding modeling and analysis. The validity and accuracy of the proposed model are determined by comparing its analysis results and simulation and measurement results.

The results show that copper windings have better thermal behavior and lower temperature compared to aluminum windings. On the other hand, by impregnating or increasing the packing factor of the winding, the thermal behavior is improved. Also, by choosing the right duty cycle for the winding current source, it is possible to prevent the burning or damage of the winding and increase its lifespan. Comparing the measurement results with the analysis results shows that the proposed equivalent circuit has an error of less than 4% in the calculation of the winding center temperature.

In this paper, the effect of temperature on the electrical resistance of the coil is ignored. Also, rectangular wires were not investigated. Research in these topics are considered as future work.

By calculating the thermal time constant of the winding, its safe operation time can be calculated so that its temperature does not exceed the tolerable value (150 °C). The proposed method analyzes both impregnated and nonimpregnated windings with various schemes. It investigates the effects of switching on their thermal behavior. Additionally, it determines the safe operating time for each type of winding.

The purpose of this paper is to solve the local problem involving strong contrast heterogeneous conductive material, with application to gas-filled porous media with both perfect and imperfect Kapitza boundary conditions at the bi-material interface. The effective parameters like the dynamic conductivity and the thermal permeability in the acoustics of porous media are also derived from the cell solution.

The Fourier transform method is used to solve frequency-dependent heat transfer problems. The periodic Lippmann–Schwinger integral equation in Fourier space with source term is first formulated using discrete Green operators and modified wavevectors, which can then be solved by iteration schemes.

Numerical examples show that the schemes converge fast and yield accurate results when compared with analytical solution for benchmark problems.

The formulation of the method is constructed using static and dynamic Green operators and can be applied to pixelized microstructure issued from tomography images.