Study on the finite element method for fluid flows with moving boundary

动边界流体流动有限元方法研究

• 批准号: 12640110
• 负责人: OHMORI Katsushi
• 金额: $ 2.3万
• 依托单位: University of Toyama
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640110/
• 关键词: Two-fluid flows; Incompressible; Immiscible; Finite element method; Convergence of interface; Mass conservation; 擬密度関数

This study has been carried out during 2000-2002 in order to develop and analyze the finite element method for fluid flows with moving boundary, which are often appeared in nature and some industrial processes.In 2000, we have considered the convergence of the approximate interface in the finite element approximation for the incompressible immiscible two-fluid flows. We have estimated the L^p (Ω)norm of the difference between the measure of the positive value of the pseudo-density function and its approximation by using the Heaviside operator H(・). Due to our result, the convergence rate is O(h^<2k/3p>) in the case of P_k-finite element. This result has been tested by numerical experiments.In 2001, we have improved the previous result by means of the regularized Heaviside operator. Then the convergence rate of the approximate interface measured by the L^2(Ω)-norm is O(h^<1/2>) in the case of P_1 or P_1isoP_2-element. This result is also tested by some numerical experiments.In 2003, we have considered the mass conservative finite element scheme for incompressible immiscible two-fluid flows. We have proposed the mixed variational formulation with the flux functional for Navier-Stokes equations. We have proved the existence and the uniqueness for continuous and approximate problems, however, the application to the non-stationary problem and the numerical computation are our theme in the near future.

本文的研究工作是在2000-2002年进行的，目的是发展和分析自然界和工业过程中经常出现的具有移动边界的流体流动的有限元方法。我们利用Heaviside算子H（·）估计了伪密度函数的正值度量与其逼近之间的差的L^p（Ω）范数。在P_k-有限元情形下，本文的结果的收敛速度为O（h^<2k/3 p>）。在2001年，我们利用正则化的Heaviside算子改进了已有的结果。在P_1或P_1isoP_2-元的情况下，用L^2（Ω）-范数度量的逼近界面的收敛速度为O（h^<1/2>）。在2003年，我们考虑了不可压不混溶两相流的质量守恒有限元格式。我们提出了Navier-Stokes方程的带通量泛函的混合变分形式。我们已经证明了连续问题和近似问题的解的存在唯一性，但在非定常问题和数值计算中的应用是我们今后的研究方向。

项目成果

Y.Fujita, K.Ohmori: "A comparison theorem for Bellman equations of ergodic control"Differential and Integral Equations. 16(to appear). (2003)

Y.Fujita、K.Ohmori：“遍历控制贝尔曼方程的比较定理”微分方程和积分方程。

H.Ikeda, M.Mimura, H.Okamoto: "A singular perturbation problem arising in Oseen's spiral flows"Japan Journal of Industrial and Applied Mathematics. 18. 393-403 (2001)

H.Ikeda、M.Mimura、H.Okamoto：“Oseen 螺旋流中出现的奇异扰动问题”日本工业与应用数学杂志。

Y.Fujita: "An auxiliary equation for the Bellman equations in a one-dimensional ergodic control"Applied Mathematics and Optimization. 43. 169-186 (2001)

Y.Fujita：“一维遍历控制中贝尔曼方程的辅助方程”应用数学和优化。

OHMORI, Katsushi: "Convergence of the interface in the finite element approximation for two-fluid flows"Lecture Notes in Pure and Applied Mathematics. 223. 279-291 (2002)

OHMORI, Katsushi：“二流体流动的有限元近似中界面的收敛”纯粹与应用数学讲义。

K.Ohmori: "Convergence of the interface in the finite element approximation for two-fluid flows"Navier-Stokes Equations : Theory & Numerical Methods (ed.by R.Salvi).. (to appear). (2001)

K.Ohmori：“两种流体流动的有限元近似中界面的收敛性”纳维-斯托克斯方程：理论