A connected component labeling algorithm is developed for implicitly defined domains specified by multivariate polynomials. The algorithm operates by recursively subdividing the constraint domain into hyperrectangular subcells until the topology thereon is sufficiently simple; in particular, we devise a topology test using properties of Bernstein polynomials. In many cases the algorithm produces a certificate guaranteeing its correctness, i.e., two points yield the same label if and only if they are path-connected. To robustly handle various kinds of edge cases, the algorithm may assign identical labels to distinct components, but only when they are exactly or nearly touching, relative to a user-controlled length scale. A variety of numerical experiments assess the effectiveness of the overall approach, including statistical analyses on randomly generated multicomponent geometry in 2D and 3D, as well as specific examples involving cusps, self-intersections, junctions, and other kinds of singularities.

With an emphasis on achieving ideal multigrid solver performance, this paper explores the design of local discontinuous Galerkin schemes for multiphase elliptic interface problems. In particular, for cases exhibiting coefficient discontinuities several orders in magnitude, the role of viscosity-weighted numerical fluxes on interfacial mesh faces is examined: findings support a known strategy of harmonic weighting, but also show that further improvements can be made via a stronger kind of biasing, denoted herein as viscosity-upwinded weighting. Applying this strategy, multigrid performance is assessed for a variety of elliptic interface problems in 1D, 2D, and 3D, across 16 orders of viscosity ratio. These include constant-and variable-coefficient problems, multiphase checkerboard patterns, implicitly defined interfaces, and 3D problems with intricate geometry. With the exception of a challenging case involving a lattice of vanishingly small droplets, in all demonstrated examples the condition number of the multigrid V-cycle preconditioned system has unit order magnitude, independent of the mesh size h.

A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly defined geometry as the graph of an implicitly defined, multi-valued height function, and applies a dimension reduction approach needing only one-dimensional quadrature. In particular, we explore the use of Gauss-Legendre and tanh-sinh methods and demonstrate that the quadrature algorithm inherits their high-order convergence rates. Under the action of h-refinement with q fixed, the quadrature schemes yield an order of accuracy of 2q, where q is the one-dimensional node count; numerical experiments demonstrate up to 22nd order. Under the action of q-refinement with the geometry fixed, the convergence is approximately exponential, i.e., doubling q approximately doubles the number of accurate digits of the computed integral. Complex geometry is automatically handled by the algorithm, including, e.g., multi-component domains, tunnels, and junctions arising from multiple polynomial level sets, as well as self-intersections, cusps, and other kinds of singularities. A variety of numerical experiments demonstrates the quadrature algorithm on two- and three-dimensional problems, including: randomly generated geometry involving multiple high-curvature pieces; challenging examples involving high degree singularities such as cusps; adaptation to simplex constraint cells in addition to hyperrectangular constraint cells; and boolean operations to compute integrals on overlapping domains.

Erdős conjectured that 1, 4, and 256 are the only powers of two whose ternary representations consist solely of 0s and 1s. Sloane conjectured that, except for {20, 21, 22, 23, 24, 215}, every other power of two has at least one 0 in its ternary represen-tation. In this paper, numerical results are given in strong support of these conjectures. In particular, we verify both conjectures for all 2n with n ≤ 2 · 345 ≈ 5.9 × 1021. Our approach makes use of a simple recursive construction of numbers 2n having prescribed patterns in their trailing ternary digits.

Modeling the physics of foams and foamlike materials, such as soapy froths, fire retardants, and lightweight crash-absorbent structures, presents challenges, because of the vastly different time and space scales involved. By separating and coupling these disparate scales, we have designed a multiscale framework to model dry foam dynamics. This leads to a predictive and flexible computational methodology linking, with a few simplifying assumptions, foam drainage, rupture, and topological rearrangement, to coupled interface-fluid motion under surface tension, gravity, and incompressible fluid dynamics. Our computed results match theoretical analyses and experimentally observed physical effects, including thin-film drainage and interference, and are used to study bubble rupture cascades and macroscopic rearrangement. The developed multiscale model allows quantitative computation of complex foam evolution phenomena.

We introduce a numerical framework, the Voronoi Implicit Interface Method for tracking multiple interacting and evolving regions (phases) whose motion is determined by complex physics (fluids, mechanics, elasticity, etc.), intricate jump conditions, internal constraints, and boundary conditions. The method works in two and three dimensions, handles tens of thousands of interfaces and separate phases, and easily and automatically handles multiple junctions, triple points, and quadruple points in two dimensions, as well as triple lines, etc., in higher dimensions. Topological changes occur naturally, with no surgery required. The method is first-order accurate at junction points/lines, and of arbitrarily high-order accuracy away from such degeneracies. The method uses a single function to describe all phases simultaneously, represented on a fixed Eulerian mesh. We test the method's accuracy through convergence tests, and demonstrate its applications to geometric flows, accurate prediction of von Neumann's law for multiphase curvature flow, and robustness under complex fluid flow with surface tension and large shearing forces.

Moving interfaces are key components of many dynamic industrial processes, in which complex interface physics determine much of the underlying action and performance. Level set methods, and their descendents, have been valuable in providing robust mathematical formulations and numerical algorithms for tracking the dynamics of these evolving interfaces. In manufacturing applications, these methods have shed light on a variety of industrial processes, including the design of industrial inkjet plotters, the mechanics of electrojetting, shape and evolution in industrial foams, and rotary bell devices in automotive painting. In this review, we discuss some of those applications, illustrating shared algorithmic challenges, and show how to tailor these methods to meet those challenges.

A mathematical model and numerical framework are presented for computing multi-physics multi-layer coating flow dynamics, with applications to the leveling of multi-layer paint films. The algorithm combines finite difference level set methods and high-order accurate sharp-interface implicit mesh discontinuous Galerkin methods to capture a complex set of multi-physics, incorporating Marangoni-driven multi-phase interfacial flow and the transport, mixing, and evaporation of multiple dissolved species. In particular, we develop several numerical methods for this multi-physics problem, including: high-order local discontinuous Galerkin methods for Poisson problems with Robin boundary conditions on implicitly-defined domains, to capture solvent evaporation; finite difference surface gradient methods, to robustly and accurately incorporate Marangoni stresses; and a coupled multi-physics time stepping approach, to incorporate all the different solvers at play including quasi-Newtonian fluid flow. The framework is applicable to an arbitrary number of layers and dissolved species; here, we apply it in a variety of settings, including multi-solvent evaporative paint dynamics, the flow and leveling of multi-layer automobile paint coatings in both 2D and 3D, and an examination of interfacial turbulence within a multi-layer matter cascade. Our results reproduce several phenomena observed in experiment, such as the formation of Marangoni plumes and Bénard cells. We also use the model to study the impact of long-wave deformational surface modes on immersed interfaces as well as the emergence of the final multi-layer film profile.

The global automotive industry sprayed over 2.6 billion liters of paint in 2018, much of which through electrostatic rotary bell atomization, a highly complex process involving the fluid mechanics of rapidly rotating thin films tearing apart into micrometer-thin filaments and droplets. Coating operations account for 65% of the energy usage in a typical automotive assembly plant, representing 10,000s of gigawatt-hours each year in the United States alone. Optimization of these processes would allow for improved robustness, reduced material waste, increased throughput, and significantly reduced energy usage. Here, we introduce a high-fidelity mathematical and algorithmic framework to analyze rotary bell atomization dynamics at industrially relevant conditions. Our approach couples laboratory experiment with the development of robust non-Newtonian fluid models; devises high-order accurate numerical methods to compute the coupled bell, paint, and gas dynamics; and efficiently exploits high-performance supercomputing architectures. These advances have yielded insight into key dynamics, including i) parametric trends in film, sheeting, and filament characteristics as a function of fluid rheology, delivery rates, and bell speed; ii) the impact of nonuniform film thicknesses on atomization performance; and iii) an understanding of spray composition via primary and secondary atomization. These findings result in coating design principles that are poised to improve energy- and cost-efficiency in a wide array of industrial and manufacturing settings.