Incompressible Fluid Application
(→ASGS(Algebraic Sub Grid Scale)) |
(→ASGS(Algebraic Sub Grid Scale)) |
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continuous solution which can not be resolved by the finite element mesh on | continuous solution which can not be resolved by the finite element mesh on | ||
the discrete finite element solution. | the discrete finite element solution. | ||
| − | + | <math>Insert formula here | |
| − | + | ||
(\rho \mbox{}\partial_{t}\mathbf{u},\mathbf{v})&+ \mu\mbox{}(\nabla\mathbf{u},\nabla\mathbf{v})+(\rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{u},\mathbf{v})-(p,\nabla\cdot\mathbf{v})+(q,\nabla\cdot\mathbf{u})\nonumber \\ | (\rho \mbox{}\partial_{t}\mathbf{u},\mathbf{v})&+ \mu\mbox{}(\nabla\mathbf{u},\nabla\mathbf{v})+(\rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{u},\mathbf{v})-(p,\nabla\cdot\mathbf{v})+(q,\nabla\cdot\mathbf{u})\nonumber \\ | ||
&+(\rho\mbox{}\partial_{t}\mathbf{u}-\mu\mbox{}\Delta\mathbf{u} + \rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{u}+\nabla p,\mu\mbox{}\Delta\mathbf{v} + \rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q)_{\tau1,t}\nonumber \\ | &+(\rho\mbox{}\partial_{t}\mathbf{u}-\mu\mbox{}\Delta\mathbf{u} + \rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{u}+\nabla p,\mu\mbox{}\Delta\mathbf{v} + \rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q)_{\tau1,t}\nonumber \\ | ||
&+(\rho\mbox{}\nabla\cdot\mathbf{u},\mbox{}\nabla\cdot\mathbf{v})_{\tau2}=\langle\mathbf{f},\mathbf{v}\rangle+(\mathbf{f},\mu\mbox{}\Delta\mathbf{v} + \rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q))_{\tau1,t} | &+(\rho\mbox{}\nabla\cdot\mathbf{u},\mbox{}\nabla\cdot\mathbf{v})_{\tau2}=\langle\mathbf{f},\mathbf{v}\rangle+(\mathbf{f},\mu\mbox{}\Delta\mathbf{v} + \rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q))_{\tau1,t} | ||
| + | </math> | ||
== Using the Application == | == Using the Application == | ||
Revision as of 16:18, 11 December 2009
Contents |
General Description
ADVERTISMENT STYLE no numerical details!!! |
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[](/images/math/d/0/b/d0b6d9ab6388ed55bb44b341fe0f5c6e.png)
![\quad \quad \quad \quad \quad \nabla\cdot\mathbf{u} = 0 \quad \text{in} \quad \Omega, ]0,T[](/images/math/4/c/8/4c8241443bc9a98e420f12b6b70063d5.png)
![\mathbf{u} = \mathbf{0} \qquad \text{in} \Gamma, t\in ]0,T[](/images/math/a/f/2/af21990dd9877e36dfb0f2ae265d88c3.png)
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 | Geometry |
|---|---|
| FractionalStep | 2D,3D Geometries |
| SubgridScale | 2D,3D Geometries |
| Fluid2DGLS_expl | 2D,3D Geometries |
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. Failed to parse (lexing error): Insert formula here (\rho \mbox{}\partial_{t}\mathbf{u},\mathbf{v})&+ \mu\mbox{}(\nabla\mathbf{u},\nabla\mathbf{v})+(\rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{u},\mathbf{v})-(p,\nabla\cdot\mathbf{v})+(q,\nabla\cdot\mathbf{u})\nonumber \\ &+(\rho\mbox{}\partial_{t}\mathbf{u}-\mu\mbox{}\Delta\mathbf{u} + \rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{u}+\nabla p,\mu\mbox{}\Delta\mathbf{v} + \rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q)_{\tau1,t}\nonumber \\ &+(\rho\mbox{}\nabla\cdot\mathbf{u},\mbox{}\nabla\cdot\mathbf{v})_{\tau2}=\langle\mathbf{f},\mathbf{v}\rangle+(\mathbf{f},\mu\mbox{}\Delta\mathbf{v} + \rho\mbox{}\mathbf{a}\cdot\nabla\mathbf{v}+\nabla q))_{\tau1,t}
