A Hybrid Immersed Boundary-Lattice Boltzmann Method for Simulation of Viscoelastic Fluid Flows Interaction with Complex Boundaries

In this study, a numerical technique based on the Lattice Boltzmann method is presented to model viscoelastic fluid interaction with complex boundaries which are commonly seen in biological systems and industrial practices. In order to accomplish numerical simulation of viscoelastic fluid flows, the Newtonian part of the momentum equations is solved by the Lattice Boltzmann Method (LBM) and the divergence of the elastic tensor, which is solved by the finite difference method, is added as a force term to the governing equations. The fluid-structure interaction forces are implemented through the Immersed Boundary Method (IBM). The numerical approach is validated for Newtonian and viscoelastic fluid flows in a straight channel, a four-roll mill geometry as well as flow over a stationary and rotating circular cylinder. Then, a numerical simulation of Oldroyd-B fluid flow around a confined elliptical cylinder with different aspect ratios is carried out for the first time. Finally, the present numerical approach is used to simulate a biological problem which is the mucociliary transport process of human respiratory system. The present numerical results are compared with appropriate analytical, numerical and experimental results obtained from the literature.     

The Immersed Interface Method for Non-Smooth Rigid Objects in Incompressible Viscous Flows

In the immersed interface method, an object in a flow is formulated as a singular force, and jump conditions caused by the singular force are incorporated into numerical schemes to compute the flow. Previous development of the method considered only smooth objects. We here extend the method to handle non-smooth rigid objects with sharp corners in 2D incompressible viscous flows. We represent the boundary of an object as a polygonal curve moving through a fixed Cartesian grid. We compute necessary jump conditions to achieve boundary condition capturing on the object. We incorporate the jump conditions into finite difference schemes to solve the flow on the Cartesian grid. The accuracy, efficiency and robustness of our method are tested using canonical flow problems. The results demonstrate that the method has second-order accuracy for the velocity and first-order accuracy for the pressure in the infinity norm, and is extremely efficient and robust to simulate flows around non-smooth complex objects.     

Effective Force Stabilising Technique for the Immersed Boundary Method

The immersed boundary method has emerged as an efficient approach for the simulation of finite-sized solid particles in complex fluid flows. However, one of the well known shortcomings of the method is the limited support for the simulation of light particles, i.e. particles with a density lower than that of the surrounding fluid, both in terms of accuracy and numerical stability.Although a broad literature exists, with several authors reporting different approaches for improving the stability of the method, most of these attempts introduce extra complexities and are very costly from a computational point of view.In this work, we introduce an effective force stabilizing technique, allowing to extend the stability range of the method by filtering spurious oscillations arising when dealing with light-particles, pushing down the particle-to-fluid density ratio as low as 0.04. We thoroughly validate the method comparing with both experimental and numerical data available in literature.     

Lattice Boltzmann Simulations of Two Linear Microswimmers Using the Immersed Boundary Method

The performance of a single or the collection of microswimmers strongly depends on the hydrodynamic coupling among their constituents and themselves. We present a numerical study for a single and a pair of microswimmers based on lattice Boltzmann method (LBM) simulations. Our numerical algorithm consists of two separable parts. Lagrange polynomials provide a discretization of the microswimmers and the lattice Boltzmann method captures the dynamics of the surrounding fluid. The two components couple via an immersed boundary method. We present data for a single swimmer system and our data also show the onset of collective effects and, in particular, an overall velocity increment of clusters of swimmers.     

Kinetic Slip Boundary Condition for Isothermal Rarefied Gas Flows Through Static Non-Planar Geometries Based on the Regularized Lattice-Boltzmann Method

The simulation of rarefied gas flows through complex porous media is challenging due to the tortuous flow pathways inherent to such structures. The Lattice Boltzmann method (LBM) has been identified as a promising avenue to solve flows through complex geometries due to the simplicity of its scheme and its high parallel computational efficiency. It has been proposed to model the stress-strain relationship with the extended Navier-Stokes equations rather than attempting to directly solve the Boltzmann equation. However, a regularization technique is required to filter out non-resolved higher-order components with a low-order velocity scheme. Although slip boundary conditions (BCs) have been proposed for the non-regularized multiple relaxation time LBM (MRT-LBM) for planar geometries, previous slip BCs have never been verified extensively with the regularization technique. In this work, following an extensive literature review on the imposition of slip BCs for rarefied flows with the LBM, it is proven that earlier values for kinetic parameters developed to impose slip BCs are inaccurate for the regularized MRT-LBM and differ between the D2Q9 and D3Q15 schemes. The error was eliminated for planar flows and good agreement between analytical solutions for arrays of cylinders and spheres was found with a wide range of Knudsen numbers.     

A Fourth-Order Kernel-Free Boundary Integral Method for Interface Problems

This paper presents a fourth-order Cartesian grid based boundary integral method (BIM) for heterogeneous interface problems in two and three dimensional space, where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions. The method reformulates the governing equation with interface conditions into boundary integral equations (BIEs) and reinterprets the involved integrals as solutions to some simple interface problems in an extended regular region. Solution of the simple equivalent interface problems for integral evaluation relies on a fourth-order finite difference method with an FFT-based fast elliptic solver. The structure of the coefficient matrix is preserved even with the existence of the interface. In the whole calculation process, analytical expressions of Green’s functions are never determined, formulated or computed. This is the novelty of the proposed kernel-free boundary integral (KFBI) method. Numerical experiments in both two and three dimensions are shown to demonstrate the algorithm efficiency and solution accuracy even for problems with a large diffusion coefficient ratio.     

A Characteristic Boundary Condition for Multispeed Lattice Boltzmann Methods

We present the development of a non-reflecting boundary condition, based on the Local One-Dimensional Inviscid (LODI) approach, for Lattice Boltzmann Models working with multi-speed stencils.We test and evaluate the LODI implementation with numerical benchmarks, showing significant accuracy gains with respect to the results produced by a simple zero-gradient condition. We also implement a simplified approach, which allows handling the unknown distribution functions spanning several layers of nodes in a unified way, still preserving a comparable level of accuracy with respect to the standard formulation.     

A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids

In this paper, a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces along the interface. The method is to apply the Taylor's expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes. A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy. The Stokes equations are discretized involving the correction terms on staggered grids and then solved by the conjugate gradient Uzawa type method. The major advantages of the present method are the special simplicity, the ability in handling the Dirichlet boundary conditions, and no need of the pressure boundary condition. The method can also preserve the volume conservation and the discrete divergence free condition very well. The numerical results show that the proposed method is second order accurate and efficient.     

Fourier Collocation and Reduced Basis Methods for Fast Modeling of Compressible Flows

A projection-based reduced order model (ROM) based on the Fourier collocation method is proposed for compressible flows. The incorporation of localized artificial viscosity model and filtering is pursued to enhance the robustness and accuracy of the ROM for shock-dominated flows. Furthermore, for Euler systems, ROMs built on the conservative and the skew-symmetric forms of the governing equation are compared. To ensure efficiency, the discrete empirical interpolation method (DEIM) is employed. An alternative reduction approach, exploring the sparsity of viscosity is also investigated for the viscous terms. A number of one- and two-dimensional benchmark cases are considered to test the performance of the proposed models. Results show that stable computations for shock-dominated cases can be achieved with ROMs built on both the conservative and the skew-symmetric forms without additional stabilization components other than the viscosity model and filtering. Under the same parameters, the skew-symmetric form shows better robustness and accuracy than its conservative counterpart, while the conservative form is superior in terms of efficiency.     

Multi-Scale Deep Neural Network (MscaleDNN) Methods for Oscillatory Stokes Flows in Complex Domains

In this paper, we study a multi-scale deep neural network (MscaleDNN) as a meshless numerical method for computing oscillatory Stokes flows in complex domains. The MscaleDNN employs a multi-scale structure in the design of its DNN using radial scalings to convert the approximation of high frequency components of the highly oscillatory Stokes solution to one of lower frequencies. The MscaleDNN solution to the Stokes problem is obtained by minimizing a loss function in terms of $L^2$ norm of the residual of the Stokes equation. Three forms of loss functions are investigated based on vorticity-velocity-pressure, velocity-stress-pressure, and velocity-gradient of velocity-pressure formulations of the Stokes equation. We first conduct a systematic study of the MscaleDNN methods with various loss functions on the Kovasznay flow in comparison with normal fully connected DNNs. Then, Stokes flows with highly oscillatory solutions in a 2-D domain with six randomly placed holes are simulated by the MscaleDNN. The results show that MscaleDNN has faster convergence and consistent error decays in the simulation of Kovasznay flow for all three tested loss functions. More importantly, the MscaleDNN is capable of learning highly oscillatory solutions when the normal DNNs fail to converge.     

Burnett Order Stress and Spatially-Dependent Boundary Conditions for the Lattice Boltzmann Method

Stress boundary conditions for the lattice Boltzmann equation that are consistent to Burnett order are proposed and imposed using a moment-based method. The accuracy of the method with complicated spatially-dependent boundary conditions for stress and velocity is investigated using the regularized lid-driven cavity flow. The complete set of boundary conditions, which involve gradients evaluated at the boundaries, are implemented locally. A recently-derived collision operator with modified equilibria and velocity-dependent collision rates to reduce the defect in Galilean invariance is also investigated. Numerical results are in excellent agreement with existing benchmark data and exhibit second-order convergence. The lattice Boltzmann stress field is studied and shown to depart significantly from the Newtonian viscous stress when the ratio of Mach to Reynolds numbers is not negligibly small.     

Adaptive Fully Implicit Simulator with Multilevel Schwarz Methods for Gas Reservoir Flows in Fractured Porous Media

Large-scale reservoir modeling and simulation of gas reservoir flows in fractured porous media is currently an important topic of interest in petroleum engineering. In this paper, the dual-porosity dual-permeability (DPDP) model coupled with the Peng-Robinson equation of state (PR-EoS) is used for the mathematical model of the gas reservoir flow in fractured porous media. We develop and study a parallel and highly scalable reservoir simulator based on an adaptive fully implicit scheme and an inexact Newton type method to solve this dual-continuum mathematical model. In the approach, an explicit-first-step, single-diagonal-coefficient, diagonally implicit Runge–Kutta (ESDIRK) method with adaptive time stepping is proposed for the fully implicit discretization, which is second-order and L-stable. And then we focus on the family of Newton–Krylov methods for the solution of a large sparse nonlinear system of equations arising at each time step. To accelerate the convergence and improve the scalability of the solver, a class of multilevel monolithic additive Schwarz methods is employed for preconditioning. Numerical results on a set of ideal as well as realistic flow problems are used to demonstrate the efficiency and the robustness of the proposed methods. Experiments on a supercomputer with several thousand processors are also carried out to show that the proposed reservoir simulator is highly scalable.     

A Simulation Approach Including Underresolved Scales for Two-Component Fluid Flows in Multiscale Porous Structures

In this study, we develop computational models and a methodology for accurate multicomponent flow simulation in underresolved multiscale porous structures [1]. It is generally impractical to fully resolve the flow in porous structures with large length-scale differences due to the tremendously high computational expense. The flow contributions from underresolved scales should be taken into account with proper physics modeling and simulation processes. Using precomputed physical properties such as the absolute permeability, $K_0,$ the capillary pressure-saturation curve, and the relative permeability, $K_r,$ in typical resolved porous structures, the local fluid force is determined and applied to simulations in the underresolved regions, which are represented by porous media. In this way, accurate flow simulations in multiscale porous structures become feasible.To evaluate the accuracy and robustness of this method, a set of benchmark test cases are simulated for both single-component and two-component flows in artificially constructed multiscale porous structures. Using comparisons with analytic solutions and results with much finer resolution resolving the porous structures, the simulated results are examined. Indeed, in all cases, the results successfully show high accuracy with proper input of $K_0,$ capillary pressure, and $K_r.$ Specifically, imbibition patterns, entry pressure, residual component patterns, and absolute/relative permeability are accurately captured with this approach.     

An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications

A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces. Since the solution has low regularity across the interface, when applying finite difference discretization to this problem, an additional treatment accounting for the jump discontinuities must be employed. Here, we aim to elevate such an extra effort to ease our implementation by machine learning methodology. The key idea is to decompose the solution into singular and regular parts. The neural network learning machinery incorporating the given jump conditions finds the singular solution, while the standard five-point Laplacian discretization is used to obtain the regular solution with associated boundary conditions. Regardless of the interface geometry, these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation, making the hybrid method easy to implement and efficient. The two- and three-dimensional numerical results show that the present hybrid method preserves second-order accuracy for the solution and its derivatives, and it is comparable with the traditional immersed interface method in the literature. As an application, we solve the Stokes equations with singular forces to demonstrate the robustness of the present method.     

Comparison of Lattice Boltzmann,Finite Element and Volume of Fluid Multicomponent Methods for Microfluidic Flow Problems and the Jetting of Microdroplets

We show that the lattice Boltzmann method (LBM) based color-gradient model with a central moments formulation (CG-CM) is capable of accurately simulating the droplet-on-demand inkjetting process on a micrometer length scale by comparing it to the Arbitrary Lagrangian Eulerian Finite Element Method (ALE-FEM). A full jetting cycle is simulated using both CG-CM and ALE-FEM and results are quantitatively compared by measuring the ejected ink velocity, volume and contraction rate. We also show that the individual relevant physical phenomena are accurately captured by considering three test-cases; droplet oscillation, ligament contraction and capillary rise. The first two cases test accuracy for a dynamic system where surface tension is the driving force and the third case is designed to test wetting boundary conditions. For the first two cases we also compare the CG-CM and ALE-FEM results to Volume of Fluid (VOF) simulations. Comparison of the three methods shows close agreement when compared to each other and analytical solutions, where available. Finally wedemonstrate that asymmetric jetting is achievable using 3D CG-CM simulations utilizing asymmetric wetting conditions inside the jet-nozzle. This allows for systematic investigation into the physics of asymmetric jetting, e.g. due to jet-nozzle manufacturing imperfections or due to other disturbances.     

Environmental Presence of Mycobacterium tuberculosis Complex in Aggregation Points at the Wildlife/Livestock Interface

The members of the Mycobacterium tuberculosis complex (MTC) cause tuberculosis (TB). Infection is transmitted within and between livestock and wildlife populations, thus hampering TB control. Indirect transmission might be facilitated if MTC bacteria persist in the environment long enough to represent a risk of exposure to different species sharing the same habitat. We have, for the first time, addressed the relationship between environmental MTC persistence and the use of water resources in two TB endemic areas in southern Spain with the objective of identifying the presence of environmental MTC and its driving factors at ungulates’ water aggregation points. Camera‐trap monitoring and MTC diagnosis (using a new MTC complex‐specific PCR technique) were carried out at watering sites. Overall, 55.8% of the water points tested positive for MTC in mud samples on the shore, while 8.9% of them were positive in the case of water samples. A higher percentage of MTC‐positive samples was found at those waterholes where cachectic animals were identified using camera‐trap monitoring, and at the smallest waterholes. Our results help to understand the role of indirect routes of cross‐species TB transmission and highlight the importance of certain environmental features in maintaining infection in multihost systems. This will help to better target actions and implement control strategies for TB at the wildlife/livestock interface.     

Molecular Hydrodynamics of the Moving Contact Line in Two-Phase Immiscible Flows

The no-slip boundary condition, i.e., zero fluid velocity relative to the solid at the fluid-solid interface, has been very successful in describing many macroscopic flows. A problem of principle arises when the no-slip boundary condition is used to model the hydrodynamics of immiscible-fluid displacement in the vicinity of the moving contact line, where the interface separating two immiscible fluids intersects the solid wall. Decades ago it was already known that the moving contact line is incompatible with the no-slip boundary condition, since the latter would imply infinite dissipation due to a non-integrable singularity in the stress near the contact line. In this paper we first present an introductory review of the problem. We then present a detailed review of our recent results on the contact-line motion in immiscible two-phase flow, from molecular dynamics (MD) simulations to continuum hydrodynamics calculations. Through extensive MD studies and detailed analysis, we have uncovered the slip boundary condition governing the moving contact line, denoted the generalized Navier boundary condition. We have used this discovery to formulate a continuum hydrodynamic model whose predictions are in remarkable quantitative agreement with the MD simulation results down to the molecular scale. These results serve to affirm the validity of the generalized Navier boundary condition, as well as to open up the possibility of continuum hydrodynamic calculations of immiscible flows that are physically meaningful at the molecular level.     

A Fourth-Order Upwinding Embedded Boundary Method (UEBM) for Maxwell's Equations in Media with Material Interfaces: Part I

In this paper, we present a new fourth-order upwinding embedded boundary method (UEBM) over Cartesian grids, originally proposed in the Journal of Computational Physics [190 (2003), pp. 159-183.] as a second-order method for treating material interfaces for Maxwell's equations. In addition to the idea of the UEBM to evolve solutions at interfaces, we utilize the ghost fluid method to construct finite difference approximation of spatial derivatives at Cartesian grid points near the material interfaces. As a result, Runge-Kutta type time discretization can be used for the semidiscretized system to yield an overall fourth-order method, in contrast to the original second-order UEBM based on a Lax-Wendroff type difference. The final scheme allows time step sizes independent of the interface locations. Numerical examples are given to demonstrate the fourth-order accuracy as well as the stability of the method. We tested the scheme for several wave problems with various material interface locations, including electromagnetic scattering of a plane wave incident on a planar boundary and a two-dimensional electromagnetic application with an interface parallel to the y-axis.     

A Highly Scalable Boundary Integral Equation and Walk-on-Spheres (BIE-WOS) Method for the Laplace Equation with Dirichlet Data

In this paper, we study a highly scalable communication-free parallel domain boundary decomposition algorithm for the Laplace equation based on a hybrid method combining boundary integral equations and walk-on-spheres (BIE-WOS) method, which provides a numerical approximation of the Dirichlet-to-Neumann (DtN) mapping for the Laplace equation. The BIE-WOS is a local method on the boundary of the domain and does not require a structured mesh, and only needs a covering of the domain boundary by patches and a local mesh for each patch for a local BIE. A new version of the BIE-WOS method with second kind integral equations is introduced for better error controls. The effect of errors from the Feynman-Kac formula based path integral WOS method on the overall accuracy of the BIE-WOS method is analyzed for the BIEs, especially in the calculation of the right hand sides of the BIEs. For the special case of flat patches, it is shown that the second kind integral equation of BIE-WOS method can be simplified where the local BIE solutions can be given in closed forms. A key advantage of the parallel BIE-WOS method is the absence of communications during the computation of the DtN mapping on individual patches of the boundary, resulting in a complete independent computation using a large number of cluster nodes. In addition, the BIE-WOS has an intrinsic capability of fault tolerance for exascale computations. The nearly linear scalability of the parallel BIE-WOS method on a large-scale cluster with 6400 CPU cores is verified for computing the DtN mapping of exterior Laplace problems with Dirichlet data for several domains.     

Weak Galerkin and Continuous Galerkin Coupled Finite Element Methods for the Stokes-Darcy Interface Problem

We consider a model of coupled free and porous media flow governed by Stokes equation and Darcy's law with the Beavers-Joseph-Saffman interface condition. In this paper, we propose a new numerical approach for the Stokes-Darcy system. The approach employs the classical finite element method for the Darcy region and the weak Galerkin finite element method for the Stokes region. We construct corresponding discrete scheme and prove its well-posedness. The estimates for the corresponding numerical approximation are derived. Finally, we present some numerical experiments to validate the efficiency of the approach for solving this problem.