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A fast variational framework for solid-fluid coupling

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Hi there, I am trying to understand the paper mentioned in the subject: http://www.cs.ubc.ca/nest/imager/tr/2007/Batty_VariationalFluids/ I am having a hard time to understand how to calculate volume fractions (or volume of fluid) for the calculations (e.g. m_i+1/2,j,k), but I am not sure I understand this part. How do I calculate the volume of fluid when a solid is cutting thru the cell? There is no information on this in the paper or the slides (although they say that it can be easily calculated from a polygon object, which I want to do, or from Dx times the area of the associated cell face, but both don´t give me a clue on how to calculate this numerically). Appreciate any help or comments! Thanks Katachi

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