Formalisme espace temps
- The space time is modeled as a pesudo-rimannian 4D manifold (link with B. Kolev - Théorie du second gradient dans le cadre relativiste - Covariance générale versus Objectivité and R. Desmorat - Sur la formulation covariante générale de l’hyperélasticité relativiste)
- Introduction of the momentum-energy tensor where is the contribution of matter to the density of energy. The idea is to do classical fluid dynamics, with the space-time formalism. First works: finite elements, finite volumes (done by Zwart et al. 1999). Interest for classical CFD
- invariance by change of frame
- same scheme in space and time
- possible to refine locally the space-time grid
- finite volumes: conservative in time also
Numerical fluid mechanics
- start from the Navier stockes equations
- When we make assumptions (no heat transfer, no viscosity, no gravity) then we obtain the Euler equations. This is a system of equations (conservation of mass, of momentum, and energy), that can be expressed as one equation, similar to an equation of conservation. He then takes this formalism with the space time formalism: this assumption makes the time derivative within the former equation of conservation to vanish !
- Then, it is possible to obtain the conservation law, expressed with the tensor momentum-energy, exactly corresponding to the conservation of mass, linear momentum, and energy.
- Example in finite difference.
- Usually, modelisation where we resolve time step by time step the whole solution in time
- In this framework, calculation of the solution at the same time !
B. Portelenelle.excalidraw
⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’
Excalidraw Data
Text Elements
time steps
spatial resolution
time
space
time
space
we can move horizontally (time) or vertically (space)
CLASSICAL 3D + 1
SPACE TIME (3+1)D
Link to original
- finite volume method
- Main idea: integration on control volumes, written in terms of fluxes with the Stockes theorem
- Interests: conservative (equality of entrance and ou), easily usable with a non structure mech.
- We have the finite volume in space, and the time integration is determine ad hoc (here it is the implicit Euler scheme = unconditionnaly stable)
- Need for the choice of a flux: here it is the Rusanov’s flux, or there exists the HLL flux
- Then, applies these ideas with the space time formalism: a space-time flux. ==Big results: able to create a space-time adaptative mesh: adaptation of the space + time mesh due to events/geometric particularities (implicit scheme so no problem)==
Remarques
- respect the principle of causality when you have a non structured space-time mesh
- conditions entropiques chocs