Smoothed Particle Hydrodynamics for Heat Transfer in Plates: Analytical Validation and Geometric Effects of Internal Heat Sources

Riski Kurniawan

doi:10.22342/jims.v32i2.2246

Riski Kurniawan ⁽¹⁾

(1) Research Center for Computing, National Research and Innovation Agency (BRIN), Indonesia

https://doi.org/10.22342/jims.v32i2.2246

Issue
Vol. 32 No. 2 (2026): JUNE

Submitted
September 23, 2025

Accepted
January 25, 2026

Published
April 23, 2026

Keywords:

Heat transfer, smoothed particle hydrodynamics, Dirichlet boundary conditions, Neumann boundary conditions, internal heat source

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Abstract

Accurate simulation of heat transfer is essential in many engineering applications. This study investigates heat transfer on thin plates by solving two-dimensional heat equations using the Smoothed Particle Hydrodynamics (SPH) method. The accuracy of the formulation is evaluated by comparing SPH results with analytical solutions for Dirichlet and Neumann boundary conditions, where small nRMSE values confirm good agreement. The method is then applied to plates with internal heat sources of different but equal-area geometries, showing that the star-shaped source yields the fastest heat transfer. These findings highlight the flexibility of SPH for modeling heat equations under complex geometries and boundary conditions.

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Authors

Riski Kurniawan

Research Center for Computing, National Research and Innovation Agency (BRIN)

riski.kurniawan@brin.go.id (Primary Contact)

Kurniawan, R. (2026). Smoothed Particle Hydrodynamics for Heat Transfer in Plates: Analytical Validation and Geometric Effects of Internal Heat Sources. Journal of the Indonesian Mathematical Society, 32(2), 2246. https://doi.org/10.22342/jims.v32i2.2246

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1. Introduction

Heat transfer is a fundamental physical process that plays a critical role in a wide range of engineering applications, from manufacturing and materials processing to energy systems and modern thermal technologies. These include heating plate panels, melting and freezing, metal casting, and thermal spraying [1,2,3,4,5]. Understanding the mechanisms of heat transfer is essential to improve energy eficiency, predict the thermal response of a system, and design devices that rely on temperature control.

Various experimental studies have been conducted to observe the conduction patterns and temperature distribution in solid materials [6,7,8]. However, experimental approaches are generally expensive and time-consuming. Therefore, mathematical models have become important alternatives for systematically studying heat transfer phenomena. In this study, the heat transfer process is modeled using the heat equation, which represents the most fundamental form of difusion[9]. Conceptually, the heat equation belongs to the family of difusion-type partial diferential equations (PDEs), including the Extended Fisher–Kolmogorov equation (EFK) [10,11] and the Kuramoto–Sivashinsky equation (KS) [12,13], which are fourth-order equations.

The analytical solution of the heat equation depends heavily on the boundary conditions and becomes intractable when the geometry or boundaries are sufficiently complex. In such cases, numerical approaches provide a more practical framework for modeling realistic heat transfer scenarios [14,15]. Classical gridbased schemes, such as the Finite Diference Method (FDM)[16], the Finite Volume Method (FVM)[17], and the Finite Element Method (FEM)[18], remain widely used for their robustness in structured domains. Various advanced discretization techniques have also been developed, including the One-Step One-Hybrid Block Method for nonlinear oscillatory PDEs[19], Hermite spline methods for the KS equation[20], and hybrid quintic Hermite spline collocation for the EFK equation[21]. Despite their strengths, grid-based methods can become restrictive when dealing with highly irregular boundaries, large deformations, or free-surface problems, motivating the use of more flexible mesh-free particle approaches such as Smoothed Particle Hydrodynamics (SPH).

SPH was first introduced by Lucy[22]and later developed independently by Gingold and Monaghan[23]for astrophysical simulations. Since then, it has been widely used to simulate solid mechanics and fluid dynamics [24,25,26]. Unlike grid-based schemes that require a predefined mesh, the SPH represents the domain as a set of interacting particles that carry quantities such as mass, position, velocity, and temperature. This mesh-free nature allows SPH to naturally handle large deformations, moving boundaries, and highly irregular geometries. The governing PDEs are reformulated as ordinary diferential equations using integral approximations based on kernels, which form the core of the SPH methodology[27].

The use of SPH for heat transfer problems began with the pioneering work of Cleary and Monaghan[28], who demonstrated its ability to solve heat conduction equations. Since then, SPH has been extensively applied to a broad range of thermal processes, including conduction, convection, and radiation in complex geometries [29,30,31,32,33,34,35]. Recent developments have further expanded its applicability to enhanced heat transfer in nanofluids[36]and chemically reactive systems where thermal difusion is coupled with reaction kinetics [37,38]. Recent advances in manufacturing further emphasize the importance of modeling internal heat sources, as demonstrated by Lin et al.[39]), who applied a ray-tracing SPH model to represent laser-induced heating. In most classical heat transfer studies, the heat source is imposed directly within the heat equation [40,41].

Although SPH has been applied in various heat transfer studies, several research gaps remain. The comprehensive validation of SPH against two-dimensional analytical solutions, especially for Dirichlet and Neumann boundary conditions, is still limited. In addition, the efect of the geometry of the internal heat source on the heat transfer rate in isolated plates, which is analytically challenging, has not been widely examined using SPH. To address these gaps, this study provides a systematic validation of the SPH against two-dimensional analytical solutions under Dirichlet and Neumann boundary conditions. Furthermore, we applied SPH to simulate heat transfer in plates containing internal heat sources of diferent geometries but equal area, allowing the assessment of how geometric variations influence heat transfer rates. The standard heat equation without a source term is employed, with the internal heat source imposed through Dirichlet conditions within the source region.

The remainder of this paper is organized as follows. Section2presents the mathematical formulation of the two-dimensional heat equation. Section3describes the SPH numerical scheme used to approximate the solution. Section4provides model validation by comparing the SPH results with analytical solutions. Section5investigates the influence of the geometry of the internal heat source on the heat transfer rate. Finally, Section6summarizes the main findings of this study.

2. Mathematical Model and Analytical Solutions

This section discusses the governing equations for heat transfer in a plate. Consider a thin plate \(\Omega \subset \mathbb { R } ^ { 2 }\) made of heat-conducting material. Let \(u ( x , y , t )\) be the temperature of the plate at point \(( x , y ) \in \Omega\) and time \(t\). This study is restricted to a rectangular plate \(\Omega = [ 0 , a ] \times [ 0

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Smoothed Particle Hydrodynamics for Heat Transfer in Plates: Analytical Validation and Geometric Effects of Internal Heat Sources

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