Simulation of heat conduction in complex domains of multi-material composites using a meshless method

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Several engineering applications involve complex materials with significant and discontinuous variations in thermophysical properties. These include materials for thermal storage, biological tissues, etc. For such applications, numerical simulations must exercise care in not smearing the interfaces by interpolating variables across the interfaces of subdomains. In this paper, we describe a high accuracy meshless method that uses domain decomposition and cloud-based interpolation of scattered data to solve heat conduction in such situations. The method is applicable to both steady state and transient problems. However, we have considered here only steady state heat conduction in complex domains. The polyharmonic spline (PHS) function with appended polynomial of prescribed degree is used for discretization. Flux balance condition must be satisfied at the interface points and the clouds of interpolation points are restricted to be within respective domains. Compared with previously proposed meshless algorithms with domain decomposition, the cloud-based interpolations are numerically better conditioned, and achieve high accuracy through the appended polynomial. The accuracy of the algorithm is demonstrated in several two and three dimensional problems using manufactured solutions with sharp discontinuity in thermal conductivity. Subsequently, we demonstrate the applicability of the algorithm to solve heat conduction in complex domains with practical boundary conditions and internal heat generation. Systematic computations with varying conductivity ratios, interpoint spacing and polynomial degree are performed to investigate the accuracy of the algorithm.