Abstract
To eliminate the numerical oscillations appearing in the first-order symmetric smoothed particle hydrodynamics (FO-SSPH) method for simulating transient heat conduction problems with discontinuous initial distribution, this paper presents a second-order symmetric smoothed particle hydrodynamics (SO-SSPH) method. Numerical properties of both SO-SSPH and FO-SSPH are analyzed, including truncation error, numerical accuracy, convergence rate, and stability. Experimental results show that for transient heat conduction with initial smooth distribution, both FO-SSPH and SO-SSPH can achieve second order convergence rate, which is consistent with the theoretical analysis. However, for one- and two-dimensional conduction with initial discontinuity, the FO-SSPH method suffers from serious unphysical oscillations, which do not disappear over time, and hence it only achieves a first-order convergence rate; while the present SO-SSPH method can avoid unphysical oscillations and has second-order convergence rate. Therefore, the SO-SSPH method is a feasible tool for solving transient heat conduction problems with both smooth and discontinuous distributions, and it is easy to be extended to high dimensional cases.
Highlights
Heat transfer is generally defined as the phenomenon where heat is spread from one system or one part to another, and can be divided into three modes: heat conduction, heat convection, and thermal radiation
The first-order symmetric smoothed particle hydrodynamics (FO-SSPH) method can more effectively improve the numerical accuracy and stability than the standard
Through theoretical analysis and numerical tests, we found that it is true for transient heat conduction problems with initial smooth distribution, but the FO-SSPH has only first-order convergence rate for problems with initial discontinuity
Summary
Heat transfer is generally defined as the phenomenon where heat is spread from one system or one part to another, and can be divided into three modes: heat conduction, heat convection, and thermal radiation. Based on Taylor series expansion both for regular and irregular particle distributions, Fatehi and Manzari [30] analyzed the truncation errors in the above three SPH approximations for second derivative and found that none of the three schemes has first-order completeness. This is because the temperature gradient at a discontinuous position tends to be infinite in theory, while the approximated first derivative in Taylor expansion can only be a finite value, so the introducing of the heat fluxes will magnify the error at this time It is difficult for FO-SSPH to solve transient heat conduction problems with initial discontinuity. Based on the above discussion, a second-order symmetric smoothed particle hydrodynamics (SO-SSPH) method (named because the Taylor series of the function is expanded up to second derivatives) is proposed to approximate second-order spatial derivatives directly in this paper to eliminate the unphysical oscillations. For both one- and two-dimensional transient heat conduction problems with a discontinuous initial distribution, the FO-SSPH method has only first-order convergence rate due to numerical oscillations, while the SO-SSPH method avoids unphysical oscillations and achieves second-order convergence
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