# Incompressible Fluid Application

(Difference between revisions)
 Revision as of 14:11, 11 December 2009 (view source)Kazem (Talk | contribs) (→Theory)← Older edit Latest revision as of 12:18, 7 June 2010 (view source)Pooyan (Talk | contribs) (→General Description) (33 intermediate revisions by 2 users not shown) Line 5: Line 5: |-style="background:#F1FAFF;" |-style="background:#F1FAFF;" | [[Image:cylinder_vel.jpg|400px]] | [[Image:cylinder_vel.jpg|400px]] − ADVERTISMENT STYLE no numerical details!!! + |-style="background:#F1FAFF;" + | + + + | An offshore platform subjected to waves. Problem solved using edgebased levelset with 3000000 elements + |} |} Line 27: Line 38: [/itex] [/itex] − This application solve the the equations.... + $− Mathematical approach to the problems. + \mathbf{u} = \mathbf{0} \qquad \text{in} \Gamma, t\in ]0,T[ +$ − Nothing numerical − Insert here all the references to your papers... + Different approaches could be chosen to solve this problem. '''Fractional step''', '''Subgrid scale stabilization''', '''GLS''' are among the others. + Some '''references''' to these methods are: − \qquad \qquad \qquad \qquad\quad \:\:\,\nabla\cdot\mathbf{\rho\mbox{}u} = 0 \qquad \text{in} \Omega,\qquad t\in ]0,T[ + 1)''Stabilized finite element approximation of transient incompressible flows using orthogonal subscales − + Ramon Codina − \qquad \qquad \qquad \qquad \qquad\quad\:\,\mathbf{u} = \mathbf{u_{0}}  \qquad \text{in} \Omega,\qquad t=0 + Computer Methods in Applied Mechanics and Engineering − + Vol. 191 (2002), 4295-4321'' − \qquad \qquad \qquad \qquad \qquad\quad\:\:\:\,\mathbf{u} = \mathbf{0}  \qquad \text{in} \Gamma,\qquad t\in ]0,T[ + === Numerical approach === === Numerical approach === Line 50: Line 61: Every physical problem is solved defining many different ingredients. Every physical problem is solved defining many different ingredients. Try to be quite schematic. Try to be quite schematic. + ==== Elements ==== + {| class="wikitable" width="100%" style="text-align:left; background:#d0d9dd; border:0px solid #e1eaee; font-size:100%; -moz-border-radius-topleft:0px; -moz-border-radius-bottomleft:0px; padding:0px 0px 0px 0px;" valign="top" + !Element + !Methodology + !Time Scheme + !Geometry + |-style="background:#F1FAFF;" + | [[FLUID]] + | [[Fractional step]] + | Forward/Backward Euler + | 2D,3D Geometries + |-style="background:#F1FAFF;" + | [[ASGS]] + | [[Variational multiscale]] + | Generalized $\alpha$ + | 2D,3D Geometries + |-style="background:#F1FAFF;" + | [[Fluid2DGLS_expl]] + | [[Least square]] + | Runge-Kutta + | 2D,3D Geometries + |} == Theory == == Theory == + + + == ASGS(Algebraic Sub Grid Scale) == + The basic idea of this method is to approximate the effect of the + continuous solution which can not be resolved by the finite element mesh on + the discrete finite element solution. + + $+ (\rho \partial_{t}\mathbf{u},\mathbf{v})+ \mu(\nabla\mathbf{u},\nabla\mathbf{v})+(\rho\mathbf{a}\cdot\nabla\mathbf{u},\mathbf{v})-(p,\nabla\cdot\mathbf{v})+(q,\nabla\cdot\mathbf{u}) +$ + + $+ +(\rho\partial_{t}\mathbf{u}-\mu\Delta\mathbf{u} + \rho\mathbf{a}\cdot\nabla\mathbf{u}+\nabla p,\mu\Delta\mathbf{v} + \rho\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q)_{\tau1,t} + +(\rho\nabla\cdot\mathbf{u},\nabla\cdot\mathbf{v})_{\tau2} +$ + + $+ =\langle\mathbf{f},\mathbf{v}\rangle+(\mathbf{f},\mu\Delta\mathbf{v} + \rho\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q))_{\tau1,t} +$ == Using the Application == == Using the Application == Line 66: Line 118: == Programming Documentation == == Programming Documentation == + + [[PureConvectionEdgeBased]] [[Category: Applications]] [[Category: Applications]]

## General Description

 An offshore platform subjected to waves. Problem solved using edgebased levelset with 3000000 elements

### Theory

The aim of this application is to solve the well known set of Navier-Stokes equations. The problem suffers from severe locking and/or instability using linear FEM.

$\partial_{t}\mathbf{u}-\nu\Delta\mathbf{u} + \mathbf{u}\cdot\nabla\mathbf{u}+\nabla p = \mathbf{f} \quad \text{in} \quad \Omega, ]0,T[$

$\quad \quad \quad \quad \quad \nabla\cdot\mathbf{u} = 0 \quad \text{in} \quad \Omega, ]0,T[$

$\mathbf{u} = \mathbf{u_{0}} \quad \text{in} \quad \Omega, t=0$

$\mathbf{u} = \mathbf{0} \qquad \text{in} \Gamma, t\in ]0,T[$

Different approaches could be chosen to solve this problem. Fractional step, Subgrid scale stabilization, GLS are among the others.

Some references to these methods are:

1)Stabilized finite element approximation of transient incompressible flows using orthogonal subscales Ramon Codina Computer Methods in Applied Mechanics and Engineering Vol. 191 (2002), 4295-4321

### Numerical approach

All numerical details here.

This is a part quite open, depending on the application we are considering.

Every physical problem is solved defining many different ingredients. Try to be quite schematic.

#### Elements

Element Methodology Time Scheme Geometry
FLUID Fractional step Forward/Backward Euler 2D,3D Geometries
ASGS Variational multiscale Generalized α 2D,3D Geometries
Fluid2DGLS_expl Least square Runge-Kutta 2D,3D Geometries

## ASGS(Algebraic Sub Grid Scale)

The basic idea of this method is to approximate the effect of the continuous solution which can not be resolved by the finite element mesh on the discrete finite element solution.

$(\rho \partial_{t}\mathbf{u},\mathbf{v})+ \mu(\nabla\mathbf{u},\nabla\mathbf{v})+(\rho\mathbf{a}\cdot\nabla\mathbf{u},\mathbf{v})-(p,\nabla\cdot\mathbf{v})+(q,\nabla\cdot\mathbf{u})$

$+(\rho\partial_{t}\mathbf{u}-\mu\Delta\mathbf{u} + \rho\mathbf{a}\cdot\nabla\mathbf{u}+\nabla p,\mu\Delta\mathbf{v} + \rho\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q)_{\tau1,t} +(\rho\nabla\cdot\mathbf{u},\nabla\cdot\mathbf{v})_{\tau2}$

$=\langle\mathbf{f},\mathbf{v}\rangle+(\mathbf{f},\mu\Delta\mathbf{v} + \rho\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q))_{\tau1,t}$