A finite element method for the three-dimensional non-steady Navier-Stokes equations

1984 ◽  
Vol 4 (4) ◽  
pp. 349-357 ◽  
Author(s):  
A. Mizukami ◽  
M. Tsuchiya
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Element Method

Pseudo Compressible Mixed Interpolation Finite Element Method for Solving Three Dimensional Navier-Stokes Equations

2013 ◽  
Author(s):  
Shrinivas G. Apte ◽  
Brian H. Dennis
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Compressibility Factor ◽  
Navier Stokes ◽  
Implicit Time ◽  
Navier Stokes Equations ◽  
Linear Solver ◽  
Element Method

A pseudo compressible finite element method for solving three dimensional incompressible Navier-Stokes equations is presented. A physical problem discretized using tetrahedral elements with linear and quadratic interpolation functions for pressure and velocity variables respectively is then marched in time by using implicit time marching scheme based on finite differencing. The possible formation of indefinite matrix due to incompressibility constraint is avoided by inserting an artificial/pseudo time dependent term (Chorin, 1974) into the continuity equation that is eliminated when steady state is reached. This definite matrix system can then be solved using standard pre-conditioners and iterative solvers. Solutions for pressure driven flows obtained using this method are validated with the ones obtained from a standard problem of flow over a cylinder and also with numerical benchmark case of a 3-D laminar flow around an obstacle. An object oriented C++ program was developed which uses exact integrals of shape functions in its calculations rather than numerical integrations. This program was tested with different values of artificial compressibility factor (β), Reynolds numbers (Re) and grid sizes (number of Elements) and time steps (dt). The effect of these parameters on the the number of linear solver iterations required for convergence is studied efficiently using the non-dimensional numbers Pseudo Compressibility Number (PCN) and Elemental Reynolds Number (ERe). Although the relationship between the linear solver performance and these two non dimensional numbers remain complicated, it is found that there exists an optimum range of PCN as a function of ERe for which the solution convergence can be obtained with the minimum number of iterations.

A finite element method for the Navier-Stokes equations in moving domain with application to hemodynamics of the left ventricle

2017 ◽  
Vol 32 (4) ◽  
Author(s):  
Alexander Danilov ◽  
Alexander Lozovskiy ◽  
Maxim Olshanskii ◽  
Yuri Vassilevski
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Left Ventricle ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Computed Tomography Images ◽  
Time Dependent Domain ◽  
Element Method

AbstractThe paper introduces a finite element method for the Navier-Stokes equations of incompressible viscous fluid in a time-dependent domain. The method is based on a quasi-Lagrangian formulation of the problem and handling the geometry in a time-explicit way. We prove that numerical solution satisfies a discrete analogue of the fundamental energy estimate. This stability estimate does not require a CFL time-step restriction. The method is further applied to simulation of a flow in a model of the left ventricle of a human heart, where the ventricle wall dynamics is reconstructed from a sequence of contrast enhanced Computed Tomography images.

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