Machine learning algorithms for three-dimensional mean-curvature computation in the level-set method

We propose a data-driven mean-curvature solver for the level-set method. This work is the natural extension to $\mathbb{R}^3$ of our two-dimensional strategy in [DOI: 10.1007/s10915-022-01952-2][1] and the hybrid inference system of [DOI: 10.1016/j.jcp.2022.111291][2]. However, in contrast to [1,2], which built resolution-dependent neural-network dictionaries, here we develop a pair of models in $\mathbb{R}^3$, regardless of the mesh size. Our feedforward networks ingest transformed level-set, gradient, and curvature data to fix numerical mean-curvature approximations selectively for interface nodes. To reduce the problem's complexity, we have used the Gaussian curvature to classify stencils and fit our models separately to non-saddle and saddle patterns. Non-saddle stencils are easier to handle because they exhibit a curvature error distribution characterized by monotonicity and symmetry. While the latter has allowed us to train only on half the mean-curvature spectrum, the former has helped us blend the data-driven and the baseline estimations seamlessly near flat regions. On the other hand, the saddle-pattern error structure is less clear; thus, we have exploited no latent information beyond what is known. In this regard, we have trained our models on not only spherical but also sinusoidal and hyperbolic paraboloidal patches. Our approach to building their data sets is systematic but gleans samples randomly while ensuring well-balancedness. We have also resorted to standardization and dimensionality reduction and integrated regularization to minimize outliers. In addition, we leverage curvature rotation/reflection invariance to improve precision at inference time. Several experiments confirm that our proposed system can yield more accurate mean-curvature estimations than modern particle-based interface reconstruction and level-set schemes around under-resolved regions

Solving elliptic problems with discontinuities on irregular domains – the Voronoi Interface Method.

We introduce a simple method, dubbed the Voronoi Interface Method, to solve Elliptic problems with discontinuities across the interface of irregular domains. This method produces a linear system that is symmetric positive definite with only its right-hand-side affected by the jump conditions. The solution and the solution's gradients are second-order accurate and first-order accurate, respectively, in the L∞L∞ norm, even in the case of large ratios in the diffusion coefficient. This approach is also applicable to arbitrary meshes. Additional degrees of freedom are placed close to the interface and a Voronoi partition centered at each of these points is used to discretize the equations in a finite volume approach. Both the locations of the additional degrees of freedom and their Voronoi discretizations are straightforward in two and three spatial dimensions

On two-phase flow solvers in irregular domains with contact line

We present numerical methods that enable the direct numerical simulation of two-phase flows in irregular domains. A method is presented to account for surface tension effects in a mesh cell containing a triple line between the liquid, gas and solid phases. Our numerical method is based on the level-set method to capture the liquid–gas interface and on the single-phase Navier–Stokes solver in irregular domain proposed in [35]to impose the solid boundary in an Eulerian framework. We also present a strategy for the implicit treatment of the viscous term and how to impose both a Neumann boundary condition and a jump condition when solving for the pressure field. Special care is given on how to take into account the contact angle, the no-slip boundary condition for the velocity field and the volume forces. Finally, we present numerical results in two and three spatial dimensions evaluating our simulations with several benchmarks

On Solving the Poisson Equation with Discontinuities on Irregular Interfaces: GFM and VIM

Publisher's version (útgefin grein)We analyze the accuracy of two numerical methods for the variable coefficient Poisson equation with discontinuities at an irregular interface. Solving the Poisson equation with discontinuities at an irregular interface is an essential part of solving many physical phenomena such as multiphase flows with and without phase change, in heat transfer, in electrokinetics, and in the modeling of biomolecules’ electrostatics. The first method, considered for the problem, is the widely known Ghost-Fluid Method (GFM) and the second method is the recently introduced Voronoi Interface Method (VIM). The VIM method uses Voronoi partitions near the interface to construct local configurations that enable the use of the Ghost-Fluid philosophy in one dimension. Both methods lead to symmetric positive definite linear systems. The Ghost-Fluid Method is generally first-order accurate, except in the case of both a constant discontinuity in the solution and a constant diffusion coefficient, while the Voronoi Interface Method is second-order accurate in the -norm. Therefore, the Voronoi Interface Method generally outweighs the Ghost-Fluid Method except in special case of both a constant discontinuity in the solution and a constant diffusion coefficient, where the Ghost-Fluid Method performs better than the Voronoi Interface Method. The paper includes numerical examples displaying this fact clearly and its findings can be used to determine which approach to choose based on the properties of the real life problem in hand.The research of Á. Helgadóttir was supported by the University of Iceland Research Fund 2015 under HI14090070. The researches of A. Guittet and F. Gibou were supported in part by the NSF under DMS-1412695 and DMREF-1534264.Peer Reviewe

Imposing mixed Dirichlet–Neumann–Robin boundary conditions in a level-set framework

Pre-print (óritrýnt handrit)We consider the Poisson equation with mixed Dirichlet, Neumann and Robin boundary conditions on irregular domains. We describe a straightforward and efficient approach for imposing the mixed boundary conditions using a hybrid finite-volume/finite-difference approach, leveraging on the work of Gibou et al. (2002) [14], Ng et al. (2009) [30] and Papac et al. (2010) [33]. We utilize three different level set functions to represent the irregular boundary at which each of the three different boundary conditions must be imposed; as a consequence, this approach can be applied to moving boundaries. The method is straightforward to implement, produces a symmetric positive definite linear system and second-order accurate solutions in the L-infinity-norm in two and three spatial dimensions. Numerical examples illustrate the second-order accuracy and the robustness of the method. (C) 2015 Elsevier Ltd. All rights reserved.The research of Á. Helgadóttir, Y.T. Ng and F. Gibou were supported in part by ONR under grant agreement N00014-11-1-0027, by the National Science Foundation under grant agreement CHE 1027817 and by the W.M. Keck Foundation. The research of C. Min was supported in part by the Kyung Hee University Research Fund (KHU-20070608) in 2007 and by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2008-331-C00045)

Laminar drag reduction in surfactant-contaminated superhydrophobic channels

While superhydrophobic surfaces (SHSs) show promise for drag reduction applications, their performance can be compromised by traces of surfactant, which generate Marangoni stresses that increase drag. This question is addressed for soluble surfactant in a three-dimensional laminar channel flow, with periodic SHSs on both walls. We assume that diffusion is sufficiently strong for cross-channel concentration gradients to be small. Exploiting a long-wave theory that accounts for a rapid transverse Marangoni-driven flow, we derive a one-dimensional model for surfactant evolution, which allows us to predict the drag reduction across the parameter space. The system exhibits multiple regimes, involving competition between Marangoni effects, bulk and interfacial diffusion, advection and shear dispersion. We map out asymptotic regions in the high-dimensional parameter space, deriving approximations of the drag reduction in each region and comparing them to numerical simulations. Our atlas of maps provides a comprehensive analytical guide for designing surfactant-contaminated channels with SHSs, to maximise the drag reduction in applications

Unsteady evolution of slip and drag in surfactant-contaminated superhydrophobic channels

Recognising that surfactants may impede the drag reduction resulting from superhydrophobic surfaces (SHSs), and that surfactant concentrations can fluctuate in space and time, we examine the unsteady transport of soluble surfactant in a laminar pressure-driven channel flow bounded between two SHSs. The SHSs are periodic in the streamwise and spanwise directions. We assume that the channel length is much longer than the streamwise period, the streamwise period is much longer than the channel height and spanwise period, and bulk diffusion is sufficiently strong for cross-channel concentration gradients to be small. By combining long-wave and homogenisation theories, we derive an unsteady advection-diffusion equation for surfactant flux transport over the length of the channel, which is coupled to a quasi-steady advection-diffusion equation for surfactant transport over individual plastrons. As diffusion over the length of the channel is typically small, the leading-order surfactant flux is governed by a nonlinear advection equation that we solve using the method of characteristics. We predict the propagation speed of a bolus of surfactant and describe its nonlinear evolution via interaction with the SHS. The propagation speed can fall significantly below the average streamwise velocity as the surfactant adsorbs and rigidifies the plastrons. Smaller concentrations of surfactant are therefore advected faster than larger ones, so that wave-steepening effects can lead to shock formation in the surfactant-flux distribution. These findings reveal the spatio-temporal evolution of the slip velocity and enable prediction of the dynamic drag reduction and effective slip length in microchannel applications

A theory for the slip and drag of superhydrophobic surfaces with surfactant.

Superhydrophobic surfaces (SHSs) have the potential to reduce drag at solid boundaries. However, multiple independent studies have recently shown that small amounts of surfactant, naturally present in the environment, can induce Marangoni forces that increase drag, at least in the laminar regime. To obtain accurate drag predictions, one must solve the mass, momentum, bulk surfactant and interfacial surfactant conservation equations. This requires expensive simulations, thus preventing surfactant from being widely considered in SHS studies. To address this issue, we propose a theory for steady, pressure-driven, laminar, two-dimensional flow in a periodic SHS channel with soluble surfactant. We linearise the coupling between flow and surfactant, under the assumption of small concentration, finding a scaling prediction for the local slip length. To obtain the drag reduction and interfacial shear, we find a series solution for the velocity field by assuming Stokes flow in the bulk and uniform interfacial shear. We find how the slip and drag depend on the nine dimensionless groups that together characterize the surfactant transport near SHSs, the gas fraction and the normalized interface length. Our model agrees with numerical simulations spanning orders of magnitude in each dimensionless group. The simulations also provide the constants in the scaling theory. Our model significantly improves predictions relative to a surfactant-free one, which can otherwise overestimate slip and underestimate drag by several orders of magnitude. Our slip length model can provide the boundary condition in other simulations, thereby accounting for surfactant effects without having to solve the full problem.Raymond and Beverly Sackler Foundation, the European Research Council Grant 247333, Mines ParisTech, the Schlumberger Chair Fund, the California NanoSystems Institute through a Challenge Grant, ARO MURI W911NF-17- 1-0306 and ONR MURI N00014-17-1-267

Slip on three-dimensional surfactant-contaminated superhydrophobic gratings

Trace amounts of surfactants have been shown to critically prevent the drag reduction of superhydrophobic surfaces (SHSs), yet predictive models including their effects in realistic geometries are still lacking. We derive theoretical predictions for the velocity and resulting slip of a laminar fluid flow over three-dimensional SHS gratings contaminated with surfactant, which allow for the first direct comparison with experiments. The results are in good agreement with our numerical simulations and with measurements of the slip in microfluidic channels lined with SHSs, which we obtain via confocal microscopy and micro-particle image velocimetry. Our model enables the estimation of a priori unknown parameters of surfactants naturally present in applications, highlighting its relevance for microfluidic technologies.Comment: 6 pages, 3 figures, 11 supplemental pages, 2 supplemental figure