Adaptive Wavelet-Galerkin Modelling of Heat Conduction in Heterogeneous Composite Materials

TL;DR

The paper tackles transient heat conduction in heterogeneous composites by introducing an adaptive wavelet-Galerkin method that combines multiresolution wavelet bases with implicit time stepping. By driving refinement with wavelet coefficient magnitudes, the method concentrates resolution near interfaces, inclusions, and boundary layers, achieving accurate solutions with substantially fewer degrees of freedom than uniform discretizations. The approach yields a sparse, compressible stiffness system and uses a preconditioned conjugate gradient solver to attain near-linear complexity as refinement grows. Numerical experiments on layered, inclusion, and functionally graded media demonstrate both accuracy and computational efficiency, highlighting the method's potential for design optimization and multiscale analysis of composite thermal transport.

Abstract

We present an adaptive wavelet Galerkin method for transient heat conduction in heterogeneous composite materials. The approach combines multiresolution wavelet bases with an implicit time discretization to efficiently resolve sharp temperature gradients near material interfaces and boundary layers. Adaptive refinement is driven by wavelet coefficients, significantly reducing the number of degrees of freedom compared to uniform discretizations. Numerical examples demonstrate accurate resolution of layered, inclusion-based, and functionally graded composites with improved computational efficiency.

Figures (3)

• Figure 1: Geometry of the two--phase layered composite slab used in Test case I. The domain $\Omega$ is split into a lower layer $\Omega_{1}$ and an upper layer $\Omega_{2}$, separated by a planar interface at $y = 1/2$ with perfect thermal contact.
• Figure 2: Geometry of the matrix--inclusion configuration used in Test case II. A circular inclusion $\Omega_{\mathrm{inc}}$ of radius $r$ is embedded in the matrix domain $\Omega_{\mathrm{m}}$, with distinct thermal conductivities in the two regions.
• Figure 3: Geometry of the Case III functionally graded material. The upper layer of the FGM $\Omega_{\mathrm{fgm}}$ is a functionally graded coating that has a gradual change in thermal conductivity, while the lower section of the material $\Omega_{\mathrm{m}}$ is a homogeneous substrate to which it is bonded.