Transient thermal analysis of a bi-layered composites with the dual-reciprocity inclusion-based boundary element method

This paper proposes a single-domain dual-reciprocity inclusion-based boundary element method (DR-iBEM) for a three-dimensional fully bonded bi-layered composite embedded with ellipsoidal inhomogeneities under transient/harmonic thermal loads. The heat equation is interpreted as a static one containing time- and frequency-dependent nonhomogeneous source terms, which is similar to eigen-fields but is transformed into a boundary integral by the dual-reciprocity method. Using the steady-state bimaterial Green’s function, boundary integral equations are proposed to take into account continuity conditions of temperature and heat flux, which avoids setting up any continuity equations at the bimaterial interface. Eigen-temperature-gradients and eigen-heat-source are introduced to simulate the material mismatch in thermal conductivity and heat capacity, respectively. The DR-iBEM algorithm is particularly suitable for investigating the transient and harmonic thermal behaviors of bi-layered composites and is verified by the finite element method (FEM). Numerical comparison with the FEM demonstrates its robustness and accuracy. The method has been applied to a functionally graded material as a bimaterial with graded particle distributions, where particle size and gradation effects are evaluated.

💡 Research Summary

The paper introduces a novel computational framework called the dual‑reciprocity inclusion‑based boundary element method (DR‑iBEM) for transient and harmonic thermal analysis of three‑dimensional fully bonded bi‑layered composites containing multiple ellipsoidal inclusions. Traditional transient Green’s function approaches are limited because they are usually expressed only in transformed (Laplace/Fourier) domains, requiring costly inverse transforms and complex interface continuity handling. DR‑iBEM overcomes these drawbacks by recasting the transient heat equation as a static problem with a time‑dependent non‑homogeneous source term, which is then treated with the dual‑reciprocity method (DRM).

The core of the method is the steady‑state bimaterial Green’s function G(x,x′) previously derived for two semi‑infinite media with different conductivities. This Green’s function inherently satisfies temperature and heat‑flux continuity across the material interface, eliminating the need for explicit interface equations. Material mismatches in thermal conductivity and specific heat are modeled by two eigen‑fields: eigen‑temperature‑gradient (ETG) and eigen‑heat‑source (EHS). Each inclusion is replaced by an “equivalent inclusion” that shares the matrix conductivity but carries the ETG and EHS to reproduce the original material contrast.

Using DRM, the domain integrals containing the non‑homogeneous source (C_p∂u/∂t) are transformed into boundary integrals by exploiting the reciprocal property of G twice. The remaining integrals over the inclusion volumes involve the static Green’s function and closed‑form Eshelby tensors, which are available analytically for ellipsoidal shapes. Consequently, no interior discretization or additional unknowns are required; only the external boundary of the composite is meshed.

Time integration is performed with a second‑order backward‑difference (Euler) scheme for transient problems, while harmonic loading is handled by separating variables (u(x,t)=ũ(x) e^{−iωt}) leading to a frequency‑domain boundary integral equation. The final linear system involves only boundary degrees of freedom, dramatically reducing the size of the algebraic problem compared with conventional multi‑domain BEM or FEM.

The authors validate DR‑iBEM against finite element simulations for both step‑change temperature loading and sinusoidal excitations over a range of frequencies. The comparison shows excellent agreement in temperature fields and heat‑flux distributions, with DR‑iBEM achieving comparable or higher accuracy while using far fewer degrees of freedom and less computational time. Notably, the method captures the correct continuity across the bimaterial interface without any special treatment.

An application to a functionally graded material (FGM) demonstrates the practical utility of the approach. By varying particle size and spatial distribution within the two layers, the study quantifies how gradation influences overall thermal response. Smaller particles produce more localized temperature perturbations, whereas higher particle volume fractions increase effective conductivity, confirming expected physical trends.

In conclusion, DR‑iBEM offers three major advantages: (1) automatic enforcement of temperature and flux continuity across material interfaces via the bimaterial Green’s function, (2) elimination of interior meshing for inclusions through the equivalent inclusion concept, and (3) the ability to treat transient and harmonic problems within a single steady‑state framework using analytical Eshelby tensors. Limitations include the current focus on linear, temperature‑independent material properties and the reliance on analytical Eshelby tensors for ellipsoidal inclusions. Future work is suggested on extending the method to nonlinear heat conduction, non‑ellipsoidal inclusions, and accelerating computations with GPU‑based implementations.