Volume 12, Issue 1
An Improved Model Reduction Method on AIMs for N-S Equations Using Multilevel Finite Element Method and Hierarchical Basis

M. Nauman Aslam, Jiazhong Zhang, Nannan Dang & Riaz Ahmad

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 115-133.

Published online: 2018-09

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  • Abstract

A numerical method is proposed to approach the Approximate Inertial Manifolds (AIMs) in unsteady incompressible Navier-Stokes equations, using multilevel finite element method with hierarchical basis functions. Following AIMS, the unknown variables, velocity and pressure in the governing equations, are divided into two components, namely low modes and high modes. Then, the couplings between low modes and high modes, which are not accounted by standard Galerkin method, are considered by AIMs, to improve the accuracy of the numerical results. Further, the multilevel finite element method with hierarchical basis functions is introduced to approach low modes and high modes in an efficient way. As an example, the flow around airfoil NACA0012 at different angles of attack has been simulated by the method presented, and the comparisons show that there is a good agreement between the present method and experimental results. In particular, the proposed method takes less computing time than the traditional method. As a conclusion, the present method is efficient in numerical analysis of fluid dynamics, especially in computing time.

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@Article{NMTMA-12-115, author = {}, title = {An Improved Model Reduction Method on AIMs for N-S Equations Using Multilevel Finite Element Method and Hierarchical Basis}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {12}, number = {1}, pages = {115--133}, abstract = {

A numerical method is proposed to approach the Approximate Inertial Manifolds (AIMs) in unsteady incompressible Navier-Stokes equations, using multilevel finite element method with hierarchical basis functions. Following AIMS, the unknown variables, velocity and pressure in the governing equations, are divided into two components, namely low modes and high modes. Then, the couplings between low modes and high modes, which are not accounted by standard Galerkin method, are considered by AIMs, to improve the accuracy of the numerical results. Further, the multilevel finite element method with hierarchical basis functions is introduced to approach low modes and high modes in an efficient way. As an example, the flow around airfoil NACA0012 at different angles of attack has been simulated by the method presented, and the comparisons show that there is a good agreement between the present method and experimental results. In particular, the proposed method takes less computing time than the traditional method. As a conclusion, the present method is efficient in numerical analysis of fluid dynamics, especially in computing time.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2017-0106}, url = {http://global-sci.org/intro/article_detail/nmtma/12693.html} }
TY - JOUR T1 - An Improved Model Reduction Method on AIMs for N-S Equations Using Multilevel Finite Element Method and Hierarchical Basis JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 115 EP - 133 PY - 2018 DA - 2018/09 SN - 12 DO - http://doi.org/10.4208/nmtma.OA-2017-0106 UR - https://global-sci.org/intro/article_detail/nmtma/12693.html KW - AB -

A numerical method is proposed to approach the Approximate Inertial Manifolds (AIMs) in unsteady incompressible Navier-Stokes equations, using multilevel finite element method with hierarchical basis functions. Following AIMS, the unknown variables, velocity and pressure in the governing equations, are divided into two components, namely low modes and high modes. Then, the couplings between low modes and high modes, which are not accounted by standard Galerkin method, are considered by AIMs, to improve the accuracy of the numerical results. Further, the multilevel finite element method with hierarchical basis functions is introduced to approach low modes and high modes in an efficient way. As an example, the flow around airfoil NACA0012 at different angles of attack has been simulated by the method presented, and the comparisons show that there is a good agreement between the present method and experimental results. In particular, the proposed method takes less computing time than the traditional method. As a conclusion, the present method is efficient in numerical analysis of fluid dynamics, especially in computing time.

M. Nauman Aslam, Jiazhong Zhang, Nannan Dang & Riaz Ahmad. (2020). An Improved Model Reduction Method on AIMs for N-S Equations Using Multilevel Finite Element Method and Hierarchical Basis. Numerical Mathematics: Theory, Methods and Applications. 12 (1). 115-133. doi:10.4208/nmtma.OA-2017-0106
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