Designing numerical methods for incompressible fluid flow involving moving interfaces, for example, in the computational modeling of bubble dynamics, swimming organisms, or surface waves, presents challenges due to the coupling of interfacial forces with incompressibility constraints. A class of methods, denoted interfacial gauge methods, is introduced for computing solutions to the corresponding incompressible Navier-Stokes equations. These methods use a type of "gauge freedom" to reduce the numerical coupling between fluid velocity, pressure, and interface position, allowing high-order accurate numerical methods to be developed more easily. Making use of an implicit mesh discontinuous Galerkin framework, developed in tandem with this work, high-order results are demonstrated, including surface tension dynamics in which fluid velocity, pressure, and interface geometry are computed with fourth-order spatial accuracy in the maximum norm. Applications are demonstrated with two-phase fluid flow displaying fine-scaled capillary wave dynamics, rigid body fluid-structure interaction, and a fluid-jet free surface flow problem exhibiting vortex shedding induced by a type of Plateau-Rayleigh instability. The developed methods can be generalized to other types of interfacial flow and facilitate precise computation of complex fluid interface phenomena.

In this two-part paper, a high-order accurate implicit mesh discontinuous Galerkin (dG) framework is developed for fluid interface dynamics, facilitating precise computation of interfacial fluid flow in evolving geometries. The framework uses implicitly defined meshes—wherein a reference quadtree or octree grid is combined with an implicit representation of evolving interfaces and moving domain boundaries—and allows physically prescribed interfacial jump conditions to be imposed or captured with high-order accuracy. Part one discusses the design of the framework, including: (i) high-order quadrature for implicitly defined elements and faces; (ii) high-order accurate discretisation of scalar and vector-valued elliptic partial differential equations with interfacial jumps in ellipticity coefficient, leading to optimal-order accuracy in the maximum norm and discrete linear systems that are symmetric positive (semi)definite; (iii) the design of incompressible fluid flow projection operators, which except for the influence of small penalty parameters, are discretely idempotent; and (iv) the design of geometric multigrid methods for elliptic interface problems on implicitly defined meshes and their use as preconditioners for the conjugate gradient method. Also discussed is a variety of aspects relating to moving interfaces, including: (v) dG discretisations of the level set method on implicitly defined meshes; (vi) transferring state between evolving implicit meshes; (vii) preserving mesh topology to accurately compute temporal derivatives; (viii) high-order accurate reinitialisation of level set functions; and (ix) the integration of adaptive mesh refinement. In part two, several applications of the implicit mesh dG framework in two and three dimensions are presented, including examples of single phase flow in nontrivial geometry, surface tension-driven two phase flow with phase-dependent fluid density and viscosity, rigid body fluid–structure interaction, and free surface flow. A class of techniques known as interfacial gauge methods is adopted to solve the corresponding incompressible Navier–Stokes equations, which, compared to archetypical projection methods, have a weaker coupling between fluid velocity, pressure, and interface position, and allow high-order accurate numerical methods to be developed more easily. Convergence analyses conducted throughout the work demonstrate high-order accuracy in the maximum norm for all of the applications considered; for example, fourth-order spatial accuracy in fluid velocity, pressure, and interface location is demonstrated for surface tension-driven two phase flow in 2D and 3D. Specific application examples include: vortex shedding in nontrivial geometry, capillary wave dynamics revealing fine-scale flow features, falling rigid bodies tumbling in unsteady flow, and free surface flow over a submersed obstacle, as well as high Reynolds number soap bubble oscillation dynamics and vortex shedding induced by a type of Plateau–Rayleigh instability in water ripple free surface flow. These last two examples compare numerical results with experimental data and serve as an additional means of validation; they also reveal physical phenomena not visible in the experiments, highlight how small-scale interfacial features develop and affect macroscopic dynamics, and demonstrate the wide range of spatial scales often at play in interfacial fluid flow.

In this two-part paper, a high-order accurate implicit mesh discontinuous Galerkin (dG) framework is developed for fluid interface dynamics, facilitating precise computation of interfacial fluid flow in evolving geometries. The framework uses implicitly defined meshes—wherein a reference quadtree or octree grid is combined with an implicit representation of evolving interfaces and moving domain boundaries—and allows physically prescribed interfacial jump conditions to be imposed or captured with high-order accuracy. Part one discusses the design of the framework, including: (i) high-order quadrature for implicitly defined elements and faces; (ii) high-order accurate discretisation of scalar and vector-valued elliptic partial differential equations with interfacial jumps in ellipticity coefficient, leading to optimal-order accuracy in the maximum norm and discrete linear systems that are symmetric positive (semi)definite; (iii) the design of incompressible fluid flow projection operators, which except for the influence of small penalty parameters, are discretely idempotent; and (iv) the design of geometric multigrid methods for elliptic interface problems on implicitly defined meshes and their use as preconditioners for the conjugate gradient method. Also discussed is a variety of aspects relating to moving interfaces, including: (v) dG discretisations of the level set method on implicitly defined meshes; (vi) transferring state between evolving implicit meshes; (vii) preserving mesh topology to accurately compute temporal derivatives; (viii) high-order accurate reinitialisation of level set functions; and (ix) the integration of adaptive mesh refinement. In part two, several applications of the implicit mesh dG framework in two and three dimensions are presented, including examples of single phase flow in nontrivial geometry, surface tension-driven two phase flow with phase-dependent fluid density and viscosity, rigid body fluid–structure interaction, and free surface flow. A class of techniques known as interfacial gauge methods is adopted to solve the corresponding incompressible Navier–Stokes equations, which, compared to archetypical projection methods, have a weaker coupling between fluid velocity, pressure, and interface position, and allow high-order accurate numerical methods to be developed more easily. Convergence analyses conducted throughout the work demonstrate high-order accuracy in the maximum norm for all of the applications considered; for example, fourth-order spatial accuracy in fluid velocity, pressure, and interface location is demonstrated for surface tension-driven two phase flow in 2D and 3D. Specific application examples include: vortex shedding in nontrivial geometry, capillary wave dynamics revealing fine-scale flow features, falling rigid bodies tumbling in unsteady flow, and free surface flow over a submersed obstacle, as well as high Reynolds number soap bubble oscillation dynamics and vortex shedding induced by a type of Plateau–Rayleigh instability in water ripple free surface flow. These last two examples compare numerical results with experimental data and serve as an additional means of validation; they also reveal physical phenomena not visible in the experiments, highlight how small-scale interfacial features develop and affect macroscopic dynamics, and demonstrate the wide range of spatial scales often at play in interfacial fluid flow.

Implicitly embedding a surface as a level set of a scalar function Φ: ℝd → ℝ is a powerful technique for computing and manipulating surface geometry. A variety of applications, e.g., level set methods for tracking evolving interfaces, require accurate approximations of minimum distances to or closest points on implicitly defined surfaces. In this paper, we present an efficient method for calculating high-order approximations of closest points on implicit surfaces, applicable to both structured and unstructured meshes in any number of spatial dimensions. In combination with a high-order approximation of Φ, the algorithm uses a rapidly converging Newton's method initialised with a guess of the closest point determined by an automatically generated point cloud approximating the surface. In general, the order of accuracy of the algorithm increases with the approximation order of Φ. We demonstrate orders of accuracy up to six for smooth problems, while nonsmooth problems reliably reduce to their expected order of accuracy. Accompanying this paper is C++ code that can be used to implement the algorithms in a variety of settings. © 2014 Mathematical Sciences Publishers.

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.

A high-order accurate implicit-mesh discontinuous Galerkin framework for wave propagation in single-phase and bi-phase solids is presented. The framework belongs to the embedded-boundary techniques and its novelty regards the spatial discretization, which enables boundary and interface conditions to be enforced with high-order accuracy on curved embedded geometries. High-order accuracy is achieved via high-order quadrature rules for implicitly-defined domains and boundaries, whilst a cell-merging strategy addresses the presence of small cut cells. The framework is used to discretize the governing equations of elastodynamics, written using a first-order hyperbolic momentum-strain formulation, and an exact Riemann solver is employed to compute the numerical flux at the interface between dissimilar materials with general anisotropic properties. The space-discretized equations are then advanced in time using explicit high-order Runge–Kutta algorithms. Several two- and three-dimensional numerical tests including dynamic adaptive mesh refinement are presented to demonstrate the high-order accuracy and the capability of the method in the elastodynamic analysis of single- and bi-phases solids containing complex geometries.

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.