Abstract The creation of a volumetric mesh representing the interior of an input polygonal mesh is a common requirement in graphics and computational mechanics applications. Most mesh creation techniques assume that the input surface is not self‐intersecting. However, due to numerical and/or user error, input surfaces are commonly self‐intersecting to some degree. The removal of self‐intersection is a burdensome task that complicates workflow and generally slows down the process of creating simulation‐ready digital assets. We present a method for the creation of a volumetric embedding hexahedron mesh from a self‐intersecting input triangle mesh. Our method is designed for efficiency by minimizing use of computationally expensive exact/adaptive precision arithmetic. Although our approach allows for nearly no limit on the degree of self‐intersection in the input surface, our focus is on efficiency in the most common case: many minimal self‐intersections. The embedding hexahedron mesh is created from a uniform background grid and consists of hexahedron elements that are geometrical copies of grid cells. Multiple copies of a single grid cell are used to resolve regions of self‐intersection/overlap. Lastly, we develop a novel topology‐aware embedding mesh coarsening technique to allow for user‐specified mesh resolution as well as a topology‐aware tetrahedralization of the hexahedron mesh.

Abstract We present a hybrid particle/grid approach for simulating incompressible fluids on collocated velocity grids. Our approach supports both particle‐based Lagrangian advection in very detailed regions of the flow and efficient Eulerian grid‐based advection in other regions of the flow. A novel Backward Semi‐Lagrangian method is derived to improve accuracy of grid based advection. Our approach utilizes the implicit formula associated with solutions of the inviscid Burgers’ equation. We solve this equation using Newton's method enabled by C1 continuous grid interpolation. We enforce incompressibility over collocated, rather than staggered grids. Our projection technique is variational and designed for B‐spline interpolation over regular grids where multiquadratic interpolation is used for velocity and multilinear interpolation for pressure. Despite our use of regular grids, we extend the variational technique to allow for cut‐cell definition of irregular flow domains for both Dirichlet and free surface boundary conditions.

We present an updated Lagrangian discretization of surface tension forces for the simulation of liquids with moderate to extreme surface tension effects. The potential energy associated with surface tension is proportional to the surface area of the liquid. We design discrete forces as gradients of this energy with respect to the motion of the fluid over a time step. We show that this naturally allows for inversion of the Hessian of the potential energy required with the use of Newton's method to solve the systems of nonlinear equations associated with implicit time stepping. The rotational invariance of the surface tension energy makes it non-convex and we define a definiteness fix procedure as in [Teran et al. 2005]. We design a novel level-set-based boundary quadrature technique to discretize the surface area calculation in our energy based formulation. Our approach works most naturally with Particle-In-Cell [Harlow 1964] techniques and we demonstrate our approach with a weakly incompressible model for liquid discretized with the Material Point Method [Sulsky et al. 1994]. We show that our approach is essential for allowing efficient implicit numerical integration in the limit of high surface tension materials like liquid metals.