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 using subdivision schemes

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

How does it improve on previous methods? • Realistic flows can be modeled and manipulated in real time

Multi-Resolution Method • Abstract: Computes a sequence of discrete approximations to solve continuous limit shape • [Insert continuous and discrete equation]

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 operator • 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

Subdivision of Cubic Splines

Fluid Mechanics • Perfect Flows: – Incompressible • Divergence is 0 – Zero Viscosity • Irrotational – Set of 2 partial differential equations

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

Finally, Subdivision of Flows • Follow the same procedure for developing subdivision of splines