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DEVELOPPEMENT DE METHODES DE VOLUMES FINIS POUR LA MECANIQUE DES FLUIDES

机译：流体力学最终体积方法的发展

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摘要

We aim to develop a finite volume method which applies to a greater class of meshes than other finite volume methods, restricted by orthogonality constraints. We build discrete differential operators over the three staggered tesselations needed for the construction of the method. These operators verify some analogous properties to those of the continuous operators. At first, the method is applied to the Div-Curl problem, which can be viewed as a building block of the Stokes problem. Then, the Stokes problem is dealt with various boundary conditions. It is well known that when the computational domain is polygonal and non-convex, the order of convergence of numerical methods is deteriored. Consequently, we have studied how an appropriate local refinement is able to restore the optimal order of convergence for the laplacian problem. At last, we have discretized the non-linear Navier-Stokes problem, using the rotational formulation of the convection term, associated to the Bernoulli pressure. With an iterative algorithm, we are led to solve a saddle--point problem at each iteration. We give a particular interest to this linear problem by testing some preconditioners issued from finite elements, which we adapt to our method. Each problem is illustrated by numerical results on arbitrary meshes, such as strongly non-conforming meshes.
机译：我们旨在开发一种有限体积方法，该方法比其他有限体积方法受正交性约束的限制，适用于更大范围的网格。我们在构造该方法所需的三个交错细分中构建离散微分算子。这些运算符验证了一些与连续运算符相似的属性。首先，该方法应用于Div-Curl问题，可以将其视为Stokes问题的基础。然后，斯托克斯问题要处理各种边界条件。众所周知，当计算域为多边形且不为凸时，数值方法的收敛顺序会变差。因此，我们研究了适当的局部细化如何能够恢复拉普拉斯问题的最优收敛阶。最后，我们使用与伯努利压力相关的对流项的旋转公式离散化了非线性Navier-Stokes问题。使用迭代算法，我们可以解决每次迭代中的鞍点问题。通过测试一些有限元发出的预处理器，我们对该方法特别感兴趣，这些预处理器适用于我们的方法。每个问题都由任意网格（例如强不合格网格）上的数值结果说明。

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    Delcourte Sarah;

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  • 年度 2007
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  • 原文格式 PDF
  • 正文语种 fr
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