The Mimetic Approach to Incompressible Surface Tension Flows

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]

Fully implicit frictional dynamics with soft constraints

Dynamics simulation with frictional contacts is important for a wide range of applications, from cloth simulation to object manipulation. Recent methods using smoothed friction forces have enabled robust and differentiable simulation of elastodynamics with friction. However, the resulting frictional behaviors can be qualitatively inaccurate and may not converge to analytic solutions. Here we propose an alternative, fully implicit, formulation for simulating elastodynamics subject to frictional contacts with realistic friction behavior. Furthermore, we demonstrate how higher-order time integration can be used in our method, as well as in incremental potential methods. We develop an inexact Newton method with forward-mode automatic differentiation that simplifies the implementation and improves performance. Finally, we show how our method can be extended to respond to volume changes using a unified penalty function derived from first principles and capable of emulating compressible as well as nearly incompressible media

Variational Stokes: A Unified Pressure-viscosity Solver for Accurate Viscous Liquids

© ACM, 2017. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in Larionov, E., Batty, C., & Bridson, R. (2017). Variational Stokes: A Unified Pressure-viscosity Solver for Accurate Viscous Liquids. ACM Trans. Graph., 36(4), 101:1–101:11. https://doi.org/10.1145/3072959.3073628We propose a novel unsteady Stokes solver for coupled viscous and pressure forces in grid-based liquid animation which yields greater accuracy and visual realism than previously achieved. Modern fluid simulators treat viscosity and pressure in separate solver stages, which reduces accuracy and yields incorrect free surface behavior. Our proposed implicit variational formulation of the Stokes problem leads to a symmetric positive definite linear system that gives properly coupled forces, provides unconditional stability, and treats difficult boundary conditions naturally through simple volume weights. Surface tension and moving solid boundaries are also easily incorporated. Qualitatively, we show that our method recovers the characteristic rope coiling instability of viscous liquids and preserves fine surface details, while previous grid-based schemes do not. Quantitatively, we demonstrate that our method is convergent through grid refinement studies on analytical problems in two dimensions. We conclude by offering practical guidelines for choosing an appropriate viscous solver, based on the scenario to be animated and the computational costs of different methods.Natural Sciences and Engineering Research Council of Canad

Constrained dynamics with frictional contact on smooth surfaces

Friction and contact pose a great challenge to efficient and accurate simulation of deformable objects for computer graphics and engineering applications. In contrast to many engineering applications, simulation software for graphics often permits larger approximation errors in favour of better predictability, controllability and efficiency. This dissertation explores modern methods for frictional contact resolution in computer graphics. In particular, the focus is on offline simulation of smooth elastic objects subject to contact with other elastic solids and cloth. We explore traditional non-smooth friction formulations as well as smoothed frictional contact, which lends itself well to differentiable simulation and analysis. We then explore a particular application of differentiable simulation to motivate the direction of research. In graphics, even smooth objects are typically approximated using piecewise linear polyhedra, which exhibit sliding artifacts that can be interpreted as artificial friction making simulations less predictable. We develop a technique for improving fidelity of sliding contact between smooth objects. Frictional contacts are traditionally resolved using non-smooth models, which are complex to analyse and difficult to compute to a desirable error estimate. We propose a unified description of the equations of motion subject to frictional contacts using a smooth model that converges to an accurate friction response. We further analyse the implications of this formulation and compare our results to state-of-the-art methods. The smooth model uniquely resolves frictional contacts, while also being fully differentiable. This allows inverse problems using our formulation to be solved by gradient-based methods. We begin our exploration of differentiable simulation applications with a parameter estimation task. Elastic parameters are estimated for a three distinct cloth materials using a novel capture, registration and estimation pipeline. Static equilibrium cloth configurations are efficiently estimated using a popular compliant constraint dynamics. In this work we address a common issue of bifurcation in cloth, which causes final configuration mismatches during estimation. Finally, we postulate an extension to compliant constraint dynamics using our friction model, to show how our previous work can be used in parameter estimation tasks involving contact and friction.Science, Faculty ofComputer Science, Department ofGraduat