# Second-Order Symmetric Smoothed Particle Hydrodynamics Method for Transient Heat Conduction Problems with Initial Discontinuity

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## Abstract

**:**

## 1. Introduction

## 2. Transient Heat Conduction Problem

## 3. SO-SSPH Method

#### 3.1. SPH Method

#### 3.2. SO-SSPH Method

## 4. Numerical Analysis

#### 4.1. Truncation Error

#### 4.2. Numerical Accuracy

#### 4.3. Convergence Rate

#### 4.4. Stability Analysis

## 5. Numerical Experiments

#### 5.1. One-Dimensional Case

#### 5.2. Two-Dimensional Case

^{2}/s, and ${T}_{0}=10\text{\hspace{0.17em}}\xb0\mathrm{C}$ are used in the simulations. The initial temperature distribution is shown in Figure 3a. Both FO-SSPH and SO-SSPH use the same simulation parameters: particle distances $\Delta x=\Delta y=0.1\text{\hspace{0.17em}}\mathrm{m}$, smoothing length $h=1.1\text{\hspace{0.17em}}\Delta x$, time step $\Delta t=0.001\text{\hspace{0.17em}}\mathrm{s}$ and simulation time 5 s.

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Abbreviations | |

CSPM | corrective smoothed particle method |

DNS | direct numerical simulation |

EFG | element-free Galerkin |

FDM | finite difference methods |

FEM | finite element methods |

FO-SSPH | first-order symmetric smoothed particle hydrodynamics |

FTCS | forward time central space |

FVM | finite volume methods |

LBM | lattice Boltzmann method |

MLSPH | moving least squares particle hydrodynamics |

MSPH | modified smoothed particle hydrodynamics |

NHT | Numerical Heat Transfer |

SO-SSPH | second-order symmetric smoothed particle hydrodynamics |

SPH | smoothed particle hydrodynamics |

Symbols | |

bT | heat source on the boundary |

c_{p} | material specific heat capacity |

E | numerical error |

erf | error function |

G | amplification factor |

G_{0} | source term |

h | smoothing length |

h | vector of particle distance |

i | imaginary unit |

k_{x} | thermal conductivity coefficients in x direction |

k_{y} | thermal conductivity coefficients in y direction |

n | unit normal vector on the boundary |

q | heat flux |

T | temperature |

T_{0} | initial temperature distribution |

T_{1} | boundary temperature |

TE | truncation error |

T^{e} | exact temperature |

T^{n} | numerical temperature |

v | column vector of the wave number in the Fourier analysis method |

W | kernel function |

W_{k} | compactly supported function |

α_{x} | coefficients of ∂f/∂x |

α_{y} | coefficients of ∂f/∂y |

β_{x} | coefficients of ∂^{2}f/∂x^{2} |

β_{y} | coefficients of ∂^{2}f/∂y^{2} |

${\mathsf{\Gamma}}_{\mathrm{D}}$ | Dirichlet boundary of the domain |

${\mathsf{\Gamma}}_{\mathrm{N}}$ | Neumann boundary of the domain |

Δt | time step |

ΔV_{j} | volume of j th particle |

Δx | particle distance in x direction |

Δy | particle distance in y direction |

κ_{x} | k_{x}/ρc_{p}, m^{2}/s |

κ_{y} | k_{y}/ρc_{p}, m^{2}/s |

ρ | material density |

Ω | computational domain |

$\langle \xb7\rangle $ | approximation operator |

$\nabla W$ | kernel gradient |

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**Figure 1.**One-dimensional heat conduction with discontinuous initial temperature distribution: (

**a**) initial condition; (

**b**) results of FO-SSPH and SO-SSPH compared with exact solution; (

**c**) SO-SSPH solutions at difference times; and (

**d**) FTCS solutions for Equation (6) (with heat fluxes) and Equation (5) (without heat fluxes).

**Figure 2.**Convergence rate of FO-SSPH and SO-SSPH for solving one-dimensional heat conduction with (

**a**) discontinuous initial distribution and (

**b**) smooth initial distribution (Gaussian).

**Figure 3.**Two-dimensional heat conduction with discontinuous temperature distribution: (

**a**) initial condition and (

**b**) results of FO-SSPH and SO-SSPH along y = 0.05 m compared with exact solution.

**Figure 4.**Temperature contours at different times: (

**a**) t = 1 s; (

**b**) t = 2 s; (

**c**) t = 3 s; (

**d**) t = 4 s; (

**e**) t = 5 s; and (

**f**) temperature distribution along y = 0.05 m compared with exact solutions.

**Figure 5.**Convergence rate of FO-SSPH and SO-SSPH for solving two-dimensional heat conduction with discontinuous initial distribution.

**Table 1.**Errors of FO-SSPH and SO-SSPH for one-dimensional heat conduction with discontinuous initial distribution.

Method | Δx (m) | |||||
---|---|---|---|---|---|---|

0.01 | 0.02 | 0.05 | 0.10 | 0.25 | 0.5 | |

FO-SSPH | 0.0089 | 0.0178 | 0.0445 | 0.0898 | 0.2297 | 0.3381 |

SO-SSPH | $5.0103\times {10}^{-5}$ | $1.9993\times {10}^{-4}$ | 0.0013 | 0.0051 | 0.0379 | 0.1303 |

**Table 2.**Errors of FO-SSPH and SO-SSPH for one-dimensional heat conduction with Gaussian distribution.

Method | Δx (m) | |||
---|---|---|---|---|

0.1 | 0.25 | 0.5 | 1.0 | |

FO-SSPH | $2.848\times {10}^{-5}$ | $1.7465\times {10}^{-4}$ | $6.8262\times {10}^{-4}$ | 0.0024 |

SO-SSPH | $7.706\text{\hspace{0.17em}}\times {10}^{-6}$ | $4.522\times {10}^{-5}$ | $1.7967\times {10}^{-4}$ | $6.5431\times {10}^{-4}$ |

**Table 3.**Errors of FO-SSPH and SO-SSPH for two-dimensional heat conduction with discontinuous initial distribution.

Method | Δx = Δy (m) | |||
---|---|---|---|---|

0.10 | 0.125 | 0.25 | 0.5 | |

FO-SSPH | 0.1548 | 0.1974 | 0.4263 | 0.6867 |

SO-SSPH | 0.0098 | 0.0159 | 0.0773 | 0.2719 |

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**MDPI and ACS Style**

Song, Z.; Xing, Y.; Hou, Q.; Lu, W. Second-Order Symmetric Smoothed Particle Hydrodynamics Method for Transient Heat Conduction Problems with Initial Discontinuity. *Processes* **2018**, *6*, 215.
https://doi.org/10.3390/pr6110215

**AMA Style**

Song Z, Xing Y, Hou Q, Lu W. Second-Order Symmetric Smoothed Particle Hydrodynamics Method for Transient Heat Conduction Problems with Initial Discontinuity. *Processes*. 2018; 6(11):215.
https://doi.org/10.3390/pr6110215

**Chicago/Turabian Style**

Song, Zhanjie, Yaxuan Xing, Qingzhi Hou, and Wenhuan Lu. 2018. "Second-Order Symmetric Smoothed Particle Hydrodynamics Method for Transient Heat Conduction Problems with Initial Discontinuity" *Processes* 6, no. 11: 215.
https://doi.org/10.3390/pr6110215