Christopher Batty

Christopher Batty, Robert Bridson

We present a simple and efficient variational finite difference method for simulating time-dependent Stokes flow in the presence of irregular free surfaces and moving solid boundaries. The method uses an embedded boundary approach on staggered Cartesian grids, avoiding the need for expensive remeshing operations, and can be applied to flows in both two and three dimensions. It uses fully implicit backwards Euler integration to provide stability and supports spatially varying density and viscosity, while requiring the solution of just a single sparse, symmetric positive-definite linear system per time step. By expressing the problem in a variational form, challenging irregular domains are supported implicitly through the use of natural boundary conditions. In practice, the discretization requires only centred finite difference stencils and per-cell volume fractions, and is straightforward to implement. The variational form further permits generalizations to coupling other mechanics, all the while reducing to a sparse symmetric positive definite matrix. We demonstrate consistent first order convergence of velocity in L1 and Linf norms on a range of analytical test cases in two dimensions. Furthermore, we apply our method as part of a simple Navier-Stokes solver to illustrate that it can reproduce the characteristic jet buckling phenomenon of Newtonian liquids at moderate viscosities, in both two and three dimensions.

Letao Chen, Sanju Mupparaju, Christopher Batty et al.

We propose a method to interpolate Signed Distance Function (SDF) data from a discrete set of samples. Unlike prior work, our approach ensures that the new SDF data values are fully consistent with the input and each other, such that the augmented data still corresponds to a geometrically realizable surface. We express the theoretical properties of SDFs as hard geometric constraints, and construct an efficient greedy algorithm for consistent SDF interpolation that is made even faster with powerful parallelized GPU preprocessing. We exemplify the usefulness of our method by evaluating it on three practical applications: global SDF refinement, in which the SDF data is upsampled without knowledge of the ground truth; mesh reconstruction, where our method can reconstruct highly detailed surfaces using global information from coarse input SDFs; and repair of pseudo-SDFs, which result from many pipelines such as CSG Boolean operations and must be turned into valid SDFs for downstream processing tasks. Our refined SDFs are guaranteed to be consistent with the input, where previous methods have no such guarantee.

Ningna Wang, Xiana Carrera, Christopher Batty et al.

Unsigned distance functions offer a powerful and flexible implicit surface representation that, unlike their signed counterparts, allow for surfaces that are open, non-orientable, or non-manifold. We consider the problem of reconstructing arbitrary surfaces from a finite set of samples of unsigned distance data. Existing methods for mesh reconstruction from distance data rely on sign information, accurate gradients, a corresponding continuous distance function, or extensive data-dependent training. However, they fail when applied to input that is both discrete and unsigned. Inspired by this challenge, we study the power diagram generated by the distance samples and propose a novel theoretical concept, the superpower contour, which we prove converges to the true surface in the limit of sampling density. We use this superpower contour as an initial surface proxy and design an algorithm that leverages it to produce a polygonal mesh approximating the unknown true geometry. Our method vastly outperforms other conceivable strategies for the discrete unsigned distance reconstruction task, and sets the stage for future work on this mathematically rich problem.

Xiana Carrera, Ningna Wang, Christopher Batty et al.

We propose an algorithm to reconstruct explicit polygonal meshes from discretely sampled Signed Distance Function (SDF) data, which is especially effective at recovering sharp features. Building on the traditional Dual Contouring of Hermite Data method, we design and solve a quadratic optimization problem to decide the optimal placement of the mesh's vertices within each cell of a regular grid. Critically, this optimization relies solely on discretely sampled SDF data, without requiring arbitrary access to the function, gradient information, or training on large-scale datasets. Our method sets a new state of the art in surface reconstruction from SDFs at medium and high resolutions, and opens the door for applications in 3D modeling and design.