"We explore methods for visually plausible fluid simulation on deforming surfaces with inhomogeneous diffusion properties. While there are methods for fluid simulation on surfaces, not much research effort focused on the influence of the motion of underlying surface, in particular when it is not a rigid surface, such as knitted or woven textiles in motion. The complexity involved makes the simulation challenging to account for the non-inertial local frames typically used to describe the motion and the anisotropic effects in diffusion, absorption, adsorption. Thus, our primary goal is to enable fast and stable method for such scenarios. First, in preparation of the material properties for the surface domain, we describe textiles with salient feature direction by bulk material property tensors in order to reduce the complexity, by employing 2D homogenization technique, which effectively turns microscale inhomogeneous properties into homogeneous properties in macroscale descriptions. We then use standard texture mapping techniques to map these tensors to triangles in the curved surface mesh, taking into account the alignment of each local tangent space with correct feature directions of the macroscale tensor. We show that this homogenization tool is intuitive, flexible and easily adjusted. Second, for efficient description of the deforming surface, we offer a new geometry representation for the surface with solely angles instead of vertex coordinates, to reduce storage for the motion of underlying surface. Since our simulation tool relies heavily on long sequences of 3D curved triangular meshes, it is worthwhile exploring such efficient representations to make our tool practical by reducing the memory access during real-time simulations as well as reducing the file sizes. Inspired by angle-based representations for tetrahedral meshes, we use spectral method to restore curved surface using both angles of the triangles and dihedral angles between adjacent triangles in the mesh. Moreover, in many surface deformation sequences, it is often sufficient to update the dihedral angles while keeping the triangle interior angles fixed. Third, we propose a framework for simulating various effects of fluid flowing on deforming surfaces. We directly applied our simulator on curved surface meshes instead of in parameter domains, whereas many existing simulation methods require a parameterization on the surface. We further demonstrate that fictitious forces induced by the surface motion can be added to the surface-based simulation at a small additional cost. These fictitious forces can be decomposed into different components. Only the rectilinear and Coriolis components are relevant to our choice of local frames. Other effects, such as diffusion, adsorption, absorption, and evaporation are also incorporated for realistic stain simulation. Finally, we explore the extraction of Lagrangian Coherent Structure (LCS), which is often referred to as the skeleton of fluid motion. The LCS structures are often described by ridges of the finite time Lyapunov exponent (FTLE) fields, which describe the extremal stretching of fluid parcels following the flow. We proposed a novel improvement to the ridge marching algorithm, which extract such ridges robustly for the typically noisy FTLE estimates even in well-defined fluid flows. Our results are potentially applicable to visualizing and controlling fluid trajectory patterns. In contrast to current methods for LCS calculation, which are only applicable to flat 2D or 3D domains and sensitive to noise, our ridge extraction is readily applicable to curved surfaces even when they are deforming. The collection of these computational tools will facilitate generation of realistic and easy to adjust surface fluid animation with various physically plausible effects on surface."--Pages ii-iii.
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