SLIDE 1
Why Subdivision of Flows?
- Fluid flow governed by non-linear partial
differential equations
- Can be simplified to linear partial
differential equations
- Flows corresponding to these linear
Subdivision of Fluid Flow Why Subdivision of Flows? Fluid flow - - PowerPoint PPT Presentation
Subdivision of Fluid Flow Why Subdivision of Flows? Fluid flow - - PowerPoint PPT Presentation
Subdivision of Fluid Flow Why Subdivision of Flows? Fluid flow governed by non-linear partial differential equations Can be simplified to linear partial differential equations Flows corresponding to these linear equations modeled
SLIDE 2
SLIDE 3
What does subdivision achieve?
- Given initial coarse vector field, generates increasingly dense
sequence of vector fields
– Limit is continuous vector field defining a flow that follows initial vector field – Follows partial differential equations
SLIDE 4
How does it improve on previous methods?
- Realistic flows can be modeled and
manipulated in real time
SLIDE 5
Multi-Resolution Method
- Abstract: Computes a sequence of
discrete approximations to solve continuous limit shape
- [Insert continuous and discrete
equation]
SLIDE 6
Multi-Resolution Method
- Multi-Grid Method:
– The domain grid T is replaced by sequence of nested grids: [Insert equation here] – D, u, and b change accordingly with T: [Insert equation here] – Use a recursive method to continually refine u
- Prediction: Compute an initial guess of the solution using a prediction
- perator
- Smoothing: Use a traditional iterative method to improve the current
solution
- Coarse grid correction: Restrict the current residual to the next
coarser grid. Solve for an error correction term and add it back to the solution
- Note that both steps 2 and 3 serve to improve the accuracy of the solution u. If
the prediction operator produces an exact initial guess then we get a SUBDIVISION SCHEME
SLIDE 7
Subdivision of Cubic Splines
SLIDE 8
Fluid Mechanics
- Perfect Flows:
– Incompressible
- Divergence is 0
– Zero Viscosity
- Irrotational
– Set of 2 partial differential equations
SLIDE 9
Primal versus Dual Subdivision
- Translating only the first component of
flow yields a new flow
- Solution: use the difference mask as
used in splines
– Yields fractional powers when m is odd (dual)
- For flows we get a hybrid
– u is primal in x and dual in y – v is primal in y and dual in x
SLIDE 10
Finally, Subdivision of Flows
- Follow the same