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.     

Hybrid Diffuse and Sharp Interface Immersed Boundary Methods for Particulate Flows in the Presence of Complex Boundaries

A coupling framework that leverages the advantages of the diffuse and sharp interface immersed boundary (IB) methods is presented for handling the interaction among particles and particles with the static complex geometries of the environment. In the proposed coupling approach, the curvilinear IB method is employed to represent the static complex geometries, a variant of the direct forcing IB method is proposed for simulating particles, and the discrete element method is employed for particle-particle and particle-wall collisions. The proposed approach is validated using several classical benchmark problems, which include flow around a sphere, sedimentation of a sphere, collision of two sedimenting spheres, and collision between a particle and a flat wall, with the present predictions showing an overall good agreement with the results reported in the literature. The capability of the proposed framework is further demonstrated by simulating the interaction between multiple particles and a wall-mounted cylinder, and the particle-laden turbulent flow over periodic hills. The proposed method provides an efficient way to simulate particle-laden turbulent flows in environments with complex boundaries.     

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 theCG-Uzawa type method and the level set representation of the interface is employedfor solving three-dimensional Stokes flow with singular forces along the interface. Themethod 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 Stokesequations are discretized involving the correction terms on staggered grids and thensolved by the conjugate gradient Uzawa type method. The major advantages of thepresent method are the special simplicity, the ability in handling the Dirichlet boundary conditions, and no need of the pressure boundary condition. The method canalso preserve the volume conservation and the discrete divergence free condition verywell. The numerical results show that the proposed method is second order accurateand efficient.     

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.     

Consistent Forcing Scheme in the Simplified Lattice Boltzmann Method for Incompressible Flows

Considering the fact that the lattice discrete effects are neglected while introducing a body force into the simplified lattice Boltzmann method (SLBM), we propose a consistent forcing scheme in SLBM for incompressible flows with external forces. The lattice discrete effects are considered at the level of distribution functions in the present forcing scheme. Consequently, it is more accurate compared with the original forcing scheme used in SLBM. Through Taylor series expansion and Chapman-Enskog (CE) expansion analysis, the present forcing scheme can be proven to recover the macroscopic Navier-Stokes (N-S) equations. Then, the macroscopic equations are resolved through a fractional step technique. Furthermore, the material derivative term is discretized by the central difference method. To verify the results of the present scheme, we simulate with multiple forms of external force interactions including the space- and time-dependent body forces. Hence, the present forcing scheme overcomes the disadvantages of the original forcing scheme and the body force can be accurately imposed in the present scheme even when a coarse mesh is applied while the original scheme fails. Excellent agreements between the analytical solutions and our numerical results can be observed.     

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.     

An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions

This paper is concerned with a novel deep learning method for variationalproblems with essential boundary conditions. To this end, we first reformulate theoriginal problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinitedimensional setting. Based on this, by expressing the primal and dual variables withtwo individual deep neural network functions, we present an augmented Lagrangiandeep learning method for which the parameters are trained by the stochastic optimization method together with a projection technique. Compared to the traditional penaltymethod, the new method admits two main advantages: i) the choice of the penaltyparameter is flexible and robust, and ii) the numerical solution is more accurate in thesame magnitude of computational cost. As typical applications, we apply the new approach to solve elliptic problems and (nonlinear) eigenvalue problems with essentialboundary conditions, and numerical experiments are presented to show the effectiveness of the new method.