The Mimetic Approach to Incompressible Surface Tension Flows
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Water has many aesthetic properties that can have a strong impact on our perceptions. For instance, coffee can bring a feeling of liveliness, rain drops on a window may spark nostalgia, morning dew on a leaf can suggest freshness, and melting icicles remind us of spring. The ubiquity of water and the complexity of phenomena it can exhibit makes it an important and interesting topic for simulation in computer graphics, where the focus lies in visual aesthetics. Our focus is to produce a method for simulating water droplets at scales where surface tension effects are visually significant. These are precisely the scales in the scenarios mentioned above. We propose a simple new approach to simulating water-like liquids with visible surface tension effects in two dimensions. We employ a recently developed Mimetic Finite Difference (MFD) method to solve the Poisson problem, which enforces incompressibility in the incompressible Euler equations. Using a cut-cell discretization, the MFD method allows us to extend the ubiquitous finite difference method on a Marker-and- Cell (MAC) discretization to handle irregular boundaries. To produce surface tension effects, we keep track of an explicit Lagrangian surface that conforms exactly to the simulation mesh. To achieve stable results, we adapt a semi-implicit surface tension scheme [Misztal et al., 2012] to our MFD pressure solve, which allows us to use time steps about 2-3 times larger than the corresponding explicit method. In addition, the semi-implicit method is extended to simulate liquids in contact with hydrophobic and hydrophilic surfaces. To provide stable surface tracking, we employ a method based on marching square templates [Rocchini et al., 2001; Müller, 2009] augmented by two simple techniques for improving mesh quality. Collapsing small interior grid edges near the fluid surface eliminates triangle elements with small angles near the liquid surface. Eliminating cells with small angles gives a better bound on the conditioning of the discrete Laplacian used in the Poisson solve, hence adding stability to our simulation. To compute surface tension forces, we use a version of the surface mesh that has been perturbed to reduce the number of short edges, in order to more robustly estimate surface curvature. These are essential for stability when coupling the MFD solve with surface tension forces. Our approach employs a unique combination of methods to address the problem of accurately tracking the contact between liquid and solid surfaces. We propose a method that couples well with the majority of fluid simulators used in the visual effects industry, while introducing a stable surface tension technique that doesn’t require complex auxiliary meshing strategies [Zheng et al., 2015].