基于有限差分法的SPH边界处理算法及其应用

doi:10.19596/j.cnki.1001-246x.8559

摘要/Abstract

摘要：

考虑虚拟粒子法在解决边界问题时因粒子搜索量大而增加冗余计算等因素，转而只在计算域内部采用SPH修正方法，边界处使用有限差分方法，从而实现一种基于有限差分法的SPH边界处理算法。以热传导问题为例，对数值模拟结果进行验证。结果表明：采用提出的边界处理方法对固定和对流边界热传导问题进行求解，解析解与数值解吻合程度较高，数值模拟可靠有效。两种算法优势互补，为相关边值问题的处理提供了思路。

关键词: 核近似, 精度, 有限差分法, 边界算法, 热传导

Abstract:

Considering influence factors of virtual particle method in solving boundary problems, such as the increase of redundant calculation due to the large amount of particle search, we adopt SPH correction method in calculation domain, and use finite difference method at boundary, to realize a SPH boundary processing algorithm based on finite difference method. Taking the heat conduction problem as an example, the numerical simulation results are verified. It shows that as the proposed boundary treatment method is used to solve fixed and convective boundary heat conduction problems, numerical solution is in good agreement with analytical solution, and the numerical simulation process is reliable and effective. Advantages of the two algorithms complement each other, which provides a new idea for treatment of related boundary value problems.

Key words: kernel approximation, accuracy, finite different method, boundary algorithms, heat conduction

裴静娴, 热合买提江·依明. 基于有限差分法的SPH边界处理算法及其应用[J]. 计算物理, 2023, 40(3): 343-352.

Jingxian PEI, Imin RAHMATJAN. SPH Boundary Algorithm Based on Finite Difference Method and Its Application[J]. Chinese Journal of Computational Physics, 2023, 40(3): 343-352.

图/表 8

图1 一元函数二阶导数计算绝对误差

Fig.1 Absolute error of second derivative calculation of unary functions

图2 二元函数二阶导数计算绝对误差(a) f1(x) = sin(πxy)，(b) f2(x) =exp(-x2-y2)，(c) f3(x) =(xy)3

Fig.2 Absolute errors of second derivative calculation of binary functions (a) f1(x) = sin(πxy), (b) f2(x) =exp(-x2-y2), (c) f3(x) =(xy)3

图3 一维固定边界条件在0.1 s，0.2 s和0.5 s时刻解析解与数值解温度曲线

Fig.3 Temperature curves of analytical and numerical solutions at 0.1 s, 0.2 s and 0.5 s for one-dimensional fixed boundary conditions

图4 一维对流边界条件在0.1 s，0.4 s和0.6 s时刻解析解与数值解温度曲线

Fig.4 Temperature curves of analytical and numerical solutions for one-dimensional convectionboundary conditions at 0.1 s, 0.4 s and 0.6 s

图5 二维固定边界条件在0.2 s，0.4 s和0.6 s时刻数值解与解析解温度曲线(a) x = 0.5, (b) y = 0.5

Fig.5 Temperature curves of numerical solutions and analytical solutions for two-dimensional fixed boundary conditions at 0.2 s, 0.4 s and 0.6 s (a)x = 0.5, (b) y = 0.5

图6 二维固定边界条件问题t=1 s时的温度分布(a)解析解，(b)数值解

Fig.6 Temperature distributions of two-dimensional fixed boundary condition problem at t=1 s (a) analytical solution, (b) numerical solution

图7 二维对流边界条件在0.2 s，0.4 s和0.6 s时刻数值解与解析解温度曲线(a)x=0.5，(b)y=0.5

Fig.7 Temperature curves of numerical solutions and analytical solutions for two-dimensional convective boundary conditions at 0.2 s, 0.4 s and 0.6 s (a)x=0.5, (b)y=0.5

图8 二维对流边界问题t = 2 s时温度分布(a)解析解，(b)数值解

Fig.8 Temperature distributions of two-dimensional convective boundary problem at t=2s (a) analytical solution, (b) numerical solution

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