Bifurcation points and bifurcated branches in fluids mechanics by high‐order mesh‐free geometric progression algorithms,International Journal for Numerical Methods in Fluids

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Bifurcation points and bifurcated branches in fluids mechanics by high‐order mesh‐free geometric progression algorithms
International Journal for Numerical Methods in Fluids ( IF 1.7 ) Pub Date : 2020-08-24 , DOI: 10.1002/fld.4910
Mohammed Rammane ₁ , Said Mesmoudi ₁ , Abdeljalil Tri _{2,

3} , Bouazza Braikat ₁ , Noureddine Damil ₁

Affiliation

In this article, we propose to investigate numerically the steady bifurcation points and bifurcated branches in fluid mechanics by employing high‐order mesh‐free geometric progression algorithms. These algorithms are based on the use of the geometric progression (GP) in a high‐order mesh‐free approach. The first proposed algorithm is applied on a strong formulation using the moving least squares (MLS) approximation coupled with GP (HO‐MLS‐GPM). While the second proposed algorithm is applied on a weak formulation using the element‐free Galerkin (EFG) coupled also with GP (HO‐EFG‐GPM). The incompressibility condition is taken by introducing the penalty technique to transform the stationary Navier–Stokes equations verified by the pressure and velocity into ones verified by only the velocity. The high‐order mesh‐free algorithm permits to transform this nonlinear equations into a succession of linear ones. The GP allows to detect with precision the bifurcation points and the Lyapunov–Schmidt reduction is coupled with HO‐MLS‐GPM and HO‐EFG‐GPM as a continuation procedure to follow the many bifurcated branches. The aim of this resolution strategy concerns the treatment of the bifurcation phenomena for a fluid flow through an expansion in several geometries, where the steady flow becomes unstable after a critical Reynolds value. The obtained results are compared with those presented in literature and with those computed using the high‐order finite element algorithm coupled with GP.

中文翻译：

高阶无网格几何级数算法在流体力学中的分支点和分支分支

在本文中，我们建议采用高阶无网格几何级数算法对流体力学中的稳态分叉点和分支分支进行数值研究。这些算法基于在高级无网格方法中使用几何级数（GP）。第一个提出的算法通过结合运动最小二乘（MLS）逼近和GP（HO-MLS-GPM）应用于强公式。虽然第二种提出的算法使用无元素Galerkin（EFG）和GP（HO-EFG-GPM）结合在弱公式上。通过引入罚分技术来将不可压缩条件转化为通过压力和速度验证的平稳Navier–Stokes方程转换为仅通过速度验证的方程。高阶无网格算法允许将该非线性方程式转换为一系列线性方程式。GP允许精确检测分叉点，而Lyapunov-Schmidt归约与HO-MLS-GPM和HO-EFG-GPM结合在一起，作为遵循许多分叉分支的延续程序。该解决方案策略的目标涉及通过几种几何形状的膨胀处理流体分叉现象的方法，在该几何形状中，经过临界的雷诺数后，稳定的流量变得不稳定。将所得结果与文献中给出的结果以及使用结合GP的高阶有限元算法计算得到的结果进行比较。GP允许精确检测分叉点，而Lyapunov-Schmidt归约与HO-MLS-GPM和HO-EFG-GPM结合在一起，作为遵循许多分叉分支的延续程序。该解决方案策略的目标涉及通过几种几何形状的膨胀处理流体分叉现象的方法，在该几何形状中，经过临界的雷诺数后，稳定的流量变得不稳定。将所得结果与文献中给出的结果以及使用结合GP的高阶有限元算法计算得到的结果进行比较。GP允许精确检测分叉点，而Lyapunov-Schmidt归约与HO-MLS-GPM和HO-EFG-GPM结合在一起，作为遵循许多分叉分支的延续程序。该解决方案策略的目标涉及通过几种几何形状的膨胀处理流体分叉现象的方法，在该几何形状中，经过临界的雷诺数后，稳定的流量变得不稳定。将所得结果与文献中给出的结果以及使用结合GP的高阶有限元算法计算得到的结果进行比较。在临界雷诺值后，稳定流量变得不稳定。将所得结果与文献中给出的结果以及使用结合GP的高阶有限元算法计算得到的结果进行比较。在临界雷诺值后，稳定流量变得不稳定。将所得结果与文献中给出的结果以及使用结合GP的高阶有限元算法计算得到的结果进行比较。

更新日期：2020-08-24

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