Heat Conduction In Heterogeneous Materials Research Articles

Implicit numerical schemes for generalized heat conduction equations

There are various situations where the classical Fourier’s law for heat conduction is not applicable, such as heat conduction in heterogeneous materials (Both et al., 2016; Ván et al., 2017) or for modeling low-temperature phenomena (Kovács and Ván, 2015, 2016, 2018). In such cases, heat flux is not directly proportional to temperature gradient, hence, the role – and both the analytical and numerical treatment – of boundary conditions becomes nontrivial. Here, we address this question for finite difference numerics via a shifted field approach. Based on this ground, implicit schemes are presented and compared to each other for the Guyer–Krumhansl generalized heat conduction equation, which successfully describes numerous beyond-Fourier experimental findings. The results are validated by an analytical solution, and are contrasted to finite element method outcomes obtained by COMSOL.

A new local meshless method for steady-state heat conduction in heterogeneous materials

In this paper a truly meshless method based on the integral form of energy equation is presented to study the steady-state heat conduction in the anisotropic and heterogeneous materials. The presented meshless method is based on the satisfaction of the integral form of energy balance equation for each sub-particle (sub-domain) inside the material. Moving least square (MLS) approximation is used for approximation of the field variable over the randomly located nodes inside the domain. In the absence of heat generation, the domain integration is eliminated from the formulation of presented method and the computational efforts are reduced substantially with respect to the conventional MLPG method. A direct method is presented for treatment of material discontinuity at the heterogeneous material in the presented meshless method. As a practical problem the heat conduction in fibrous composite material is studied and the steady-state heat conduction in unidirectional fiber–matrix composites is investigated. The solution domain includes a small area of the composite system called representative volume element (RVE). Comparison of numerical results shows that the presented meshless method is simple, effective, accurate and less costly method for micromechanical analysis of heat conduction in heterogeneous materials.

On Parallel Nonequilibrium Molecular Dynamics Simulations of Heat Conduction in Heterogeneous Materials with Three-Body Potentials: Si/Ge Superlattice

Parallelization strategies for nonequilibrium molecular dynamics (NEMD) simulations of heat conduction in heterogeneous materials are presented. In particular, a previously published algorithm involving the pair decomposition of three-body potentials is extended for heterogeneous materials. In addition, a novel and linear scaling scheme, also based on pair decomposition of three-body terms, is introduced for the calculation of the heat flux. The distributed-computing-based implementation of this algorithm is outlined and its speed-up characteristics are demonstrated to be close to ideal. Example NEMD simulations using the new algorithm are performed for the Si/Ge superlattice based on the three-body Stillinger-Weber potential.

Heat conduction in heterogeneous materials and irregular boundaries III: A computer graphics based algorithm for calculating effective thermal parameters

In earlier papers [1–3] a simple and computationally fast method for investigating heat conduction in materials of irregular shapes with heterogeneous thermal conductivity was presented. In that method, dubbed EPIC, the space is tesselated into cubical cell, an interfacial cells with regions of different thermal conductivities inside it is replaced by a cell with an effective uniform conductivity, and the heat conduction equation is solved numerically by using a finite difference method. In this paper, we present a novel computer graphics based approach to calculate effective conductivity which allows EPIC to be used for complex engineering systems generated by using standard CAD/CAM packages, bringing EPIC one step closer to becoming a practical tool.

Heat conduction in heterogeneous materials and irregular boundaries II: A simple and fast algorithm for massively parallel computer

In an earlier paper [1], a simple model for investigating heat conduction in materials of irregular shapes having heterogeneous thermal conductivity was presented. In this method, referred to as the Effective Parameter for Interfacial cells (dubbed as EPIC) [2], the material is tessellated into cells (square or cubes), a cell with non-uniform thermal conductivity inside it is replaced by a cell with an effective uniform conductivity, and the heat conduction equation is then solved numerically by discretizing in time and space. The implementation of EPIC on a massively parallel SIMD machine (MasParMP-1) is described. Sample results are shown for 2-D and 3-D materials.

Effective macroscopic description for heat conduction in heterogeneous materials

The macroscopic thermal behaviour of heterogeneous materials is studied using the ensemble averaging technique. The non-local constitutive relations for heat conduction are derived. They relate the ensemble averaged heat flux and energy density to the ensemble averaged temperature of the medium. All the effective properties appearing in the relations are defined with help of the newly introduced, so-called microstructure functions. The asymptotic behaviour of the heterogeneous media for slowly varying average temperature fields was investigated. Possible applications of the theory are presented in two examples when analytical results could be easily obtained.

A simple model for heat conduction in heterogeneous materials and irregular boundaries

A simple method for investigating heat conduction in materials of non-uniform thermal conductivity and irregular shapes is presented. In this method, the material is tessellated into cells (squares in two-dimensions and cubes in three-dimensions), the regions of non-uniform conductivities in each of the cells are replaced by the cell of an effective thermal conductivity, and the heat conduction equation is then solved numerically by discretizing it in time and space. Sample results based on this approach are given for 2-D materials.

Heat Conduction in Heterogeneous Materials

A new solution to the heat equation in composite media is derived using a variational principle developed by Ben-Amoz. The model microstructure is fed into the equations via a term for the polar moment of the inclusions in a representative volume. The general solution is presented as an integral in terms of sources and a Green function. The problem of uniqueness is studied to determine appropriate boundary conditions. The solution reduces to the solution of the normal heat equation in the limit of homogeneous media.