Subdivision of Fluid FlowWhy 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 schemesWhat 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 equationsHow does it improve on previous methods?• Realistic flows can be modeled and manipulated in real timeMulti-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 SCHEMESubdivision of Cubic SplinesFluid Mechanics• Perfect Flows:– Incompressible• Divergence is 0– Zero Viscosity• Irrotational– Set of 2 partial differential equationsPrimal 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 xFinally, Subdivision of Flows• Follow the same procedure for developing subdivision of
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