Lec FV2/FE1
20 important questions on Lec FV2/FE1
What is the great advantage of choosing CV faces centered between nodes over centering nodes in CVs
Explain the integral form of the advection/diffusion equation from the FV perspectve
It is a math theorem that says that if we want to find the change of the state variable within volume is kinda the integral of all the fluxes over the surfaces of the cube (volume). (Or find change of volume by fluxes that act on it.)
What is the difference between Q and q?
Q: flux*area (kg/s). Thus the flux multiplied by the area over which it travels
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Explain how voronoi cells are made?
Why do we have an exact mass conservation in FV?
Why do we need a different method to solve our equation instead of finding your perfect solution like on slide 9?
How does Newton Raphson work if a solution of =0 should be found?
- Function to solve is drawn, this is the balance for your CVs, which should add up to 0 when solved
- Calculate balance and slope in balance (derivative)
- Where extrapolation of derivative intersects the x-axis (=0 line), the balance is calculated in the next iteration. If solution is >0, derivative in negative direction is followed and vice versa.
- Final solution is approached by various iterations. Solution of =0 should be found, if not we have an error
How do we deal with transient equations in FV? Transient equations monitor how the balance
Balance eq for CV, we could also formulate the amount of balance that we have as smt that changes over time. So we want to track how it moves to the steady state solution over time. So weintroduce initial conditions
Initial conditions are introduced, eg H1=H2=H3 = 5 m So start flat, turn on the rain and then see what happens to states.
Why should we solve to =0 in numerical solvers?
Mention 1 general problem related to derivatives and 4 problems that could occur with the Newton Raphson method?
Problem with NR for complex (non)linear functions:
- send solver in the wrong direction
- no solution present
- local min/max could occur
- non-linear solution does not converge
How can you visually check if you have an error in Newton Raphson?
Why do we generally use implicit methods to solve transient systems?
How does FE differ from FV?
- same value assigned to entire volume, where the node is in the middle.
- You need a different shape for a different mesh
- Voronoi cells
- nodes are placed on edges/borders of triangles
Y - You can always make 2 types of grids/meshes with the same nodes.
- More smooth result
- Delauney triangularisation
- Works based on inverse distance
Voronoi and Delauney are each other's duals in mesh
How does FE work in Liekes lectures' example?
- If Q would fall on node Xleft, I assign Xleft value 1.
- If it falls on the neighbouring node (Xright), I assign 0 to node Xleft.
- In the middle, we linearly interpolate.
- If it would fall in the middle, 50/50 would be assigned to both nodes.
- If it falls off the middle, the linear interpolation determines the fraction each node gets
Does external point flux move to nodes other than the two it falls within?
Why is FE more smooth?
! Note that even though sometimes if a point flux is closer to a node that is not a neighbour, it is still divided over the two neighbours. If you want it to be more accurate, more nodes should be added. Note that this only works if the resolution of your observations is finer than your nodal distance.
How do you use FE for spatial fluxes
With integration, we take an infetisemal small step and for each step we multiply the redistribution factor with the spatial flux at that step. We do this over the complete distance in between the nodes.
External point flux in 2D
Then we can determine the total probability redistributing this external factor to a certain point.
Why do you need two integrals for the spatial flux in 2D for FE?
When does the advantage of FE really come in?
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