Strongly coupled peridynamic and lattice Boltzmann models using immersed boundary method for flow-induced structural deformation and fracture

To simulate the dynamics of structural deformation and fracture caused by fluid-structure interactions accurately and efficiently, a strong coupling between the peridynamic model and the lattice Boltzmann method using the immersed boundary method is developed here. In this novel method, the peridynamic model predicts structural deformation and fracture, the cascaded lattice Boltzmann method serves as the flow solver, and the immersed boundary method is to enforce a no-slip boundary condition on the fluid-solid interface. The strong coupling is achieved by adding velocity corrections for the fluid and solid phases simultaneously at each time step, which are calculated by solving a linear system of equations derived from an implicit velocity correction immersed boundary scheme. Therefore, this new scheme based on the immersed boundary method eliminates the need to iteratively solve the dynamics of the fluid and solid phases at each time step. The proposed method is rigorously validated considering the plate with a pre-existing crack under velocity boundary conditions, the sedimentation of an elastic disk, the cross-flow over a flexible beam, and the flow-induced deformation of an elastic beam attached to a rigid cylinder. More importantly, the structural deformation, crack formation, and fracture due to interaction with the fluid flow are captured innovatively

A unified operator splitting approach for multi-scale fluid-particle coupling in the lattice Boltzmann method

A unified framework to derive discrete time-marching schemes for coupling of immersed solid and elastic objects to the lattice Boltzmann method is presented. Based on operator splitting for the discrete Boltzmann equation, second-order time-accurate schemes for the immersed boundary method, viscous force coupling and external boundary force are derived. Furthermore, a modified formulation of the external boundary force is introduced that leads to a more accurate no-slip boundary condition. The derivation also reveals that the coupling methods can be cast into a unified form, and that the immersed boundary method can be interpreted as the limit of force coupling for vanishing particle mass. In practice, the ratio between fluid and particle mass determines the strength of the force transfer in the coupling. The integration schemes formally improve the accuracy of first-order algorithms that are commonly employed when coupling immersed objects to a lattice Boltzmann fluid. It is anticipated that they will also lead to superior long-time stability in simulations of complex fluids with multiple scales

Bridging the computational gap between mesoscopic and continuum modeling of red blood cells for fully resolved blood flow

We present a computational framework for the simulation of blood flow with fully resolved red blood cells (RBCs) using a modular approach that consists of a lattice Boltzmann solver for the blood plasma, a novel finite element based solver for the deformable bodies and an immersed boundary method for the fluid-solid interaction. For the RBCs, we propose a nodal projective FEM (npFEM) solver which has theoretical advantages over the more commonly used mass-spring systems (mesoscopic modeling), such as an unconditional stability, versatile material expressivity, and one set of parameters to fully describe the behavior of the body at any mesh resolution. At the same time, the method is substantially faster than other FEM solvers proposed in this field, and has an efficiency that is comparable to the one of mesoscopic models. At its core, the solver uses specially defined potential energies, and builds upon them a fast iterative procedure based on quasi-Newton techniques. For a known material, our solver has only one free parameter that demands tuning, related to the body viscoelasticity. In contrast, state-of-the-art solvers for deformable bodies have more free parameters, and the calibration of the models demands special assumptions regarding the mesh topology, which restrict their generality and mesh independence. We propose as well a modification to the potential energy proposed by Skalak et al. 1973 for the red blood cell membrane, which enhances the strain hardening behavior at higher deformations. Our viscoelastic model for the red blood cell, while simple enough and applicable to any kind of solver as a post-convergence step, can capture accurately the characteristic recovery time and tank-treading frequencies. The framework is validated using experimental data, and it proves to be scalable for multiple deformable bodies

Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method

The deformation of an initially spherical capsule, freely suspended in simple shear flow, can be computed analytically in the limit of small deformations [D. Barthes-Biesel, J. M. Rallison, The Time-Dependent Deformation of a Capsule Freely Suspended in a Linear Shear Flow, J. Fluid Mech. 113 (1981) 251-267]. Those analytic approximations are used to study the influence of the mesh tessellation method, the spatial resolution, and the discrete delta function of the immersed boundary method on the numerical results obtained by a coupled immersed boundary lattice Boltzmann finite element method. For the description of the capsule membrane, a finite element method and the Skalak constitutive model [R. Skalak et al., Strain Energy Function of Red Blood Cell Membranes, Biophys. J. 13 (1973) 245-264] have been employed. Our primary goal is the investigation of the presented model for small resolutions to provide a sound basis for efficient but accurate simulations of multiple deformable particles immersed in a fluid. We come to the conclusion that details of the membrane mesh, as tessellation method and resolution, play only a minor role. The hydrodynamic resolution, i.e., the width of the discrete delta function, can significantly influence the accuracy of the simulations. The discretization of the delta function introduces an artificial length scale, which effectively changes the radius and the deformability of the capsule. We discuss possibilities of reducing the computing time of simulations of deformable objects immersed in a fluid while maintaining high accuracy.Comment: 23 pages, 14 figures, 3 table

Inertial Coupling Method for particles in an incompressible fluctuating fluid

We develop an inertial coupling method for modeling the dynamics of point-like 'blob' particles immersed in an incompressible fluid, generalizing previous work for compressible fluids. The coupling consistently includes excess (positive or negative) inertia of the particles relative to the displaced fluid, and accounts for thermal fluctuations in the fluid momentum equation. The coupling between the fluid and the blob is based on a no-slip constraint equating the particle velocity with the local average of the fluid velocity, and conserves momentum and energy. We demonstrate that the formulation obeys a fluctuation-dissipation balance, owing to the non-dissipative nature of the no-slip coupling. We develop a spatio-temporal discretization that preserves, as best as possible, these properties of the continuum formulation. In the spatial discretization, the local averaging and spreading operations are accomplished using compact kernels commonly used in immersed boundary methods. We find that the special properties of these kernels make the discrete blob a particle with surprisingly physically-consistent volume, mass, and hydrodynamic properties. We develop a second-order semi-implicit temporal integrator that maintains discrete fluctuation-dissipation balance, and is not limited in stability by viscosity. Furthermore, the temporal scheme requires only constant-coefficient Poisson and Helmholtz linear solvers, enabling a very efficient and simple FFT-based implementation on GPUs. We numerically investigate the performance of the method on several standard test problems...Comment: Contains a number of corrections and an additional Figure 7 (and associated discussion) relative to published versio

Entropic Lattice Boltzmann Method for Moving and Deforming Geometries in Three Dimensions

Entropic lattice Boltzmann methods have been developed to alleviate intrinsic stability issues of lattice Boltzmann models for under-resolved simulations. Its reliability in combination with moving objects was established for various laminar benchmark flows in two dimensions in our previous work Dorschner et al. [11] as well as for three dimensional one-way coupled simulations of engine-type geometries in Dorschner et al. [12] for flat moving walls. The present contribution aims to fully exploit the advantages of entropic lattice Boltzmann models in terms of stability and accuracy and extends the methodology to three-dimensional cases including two-way coupling between fluid and structure, turbulence and deformable meshes. To cover this wide range of applications, the classical benchmark of a sedimenting sphere is chosen first to validate the general two-way coupling algorithm. Increasing the complexity, we subsequently consider the simulation of a plunging SD7003 airfoil at a Reynolds number of Re = 40000 and finally, to access the model's performance for deforming meshes, we conduct a two-way coupled simulation of a self-propelled anguilliform swimmer. These simulations confirm the viability of the new fluid-structure interaction lattice Boltzmann algorithm to simulate flows of engineering relevance.Comment: submitted to Journal of Computational Physic

An IB Method for Non-Newtonian-Fluid Flexible-Structure Interactions in Three-Dimensions

Problems involving fluid flexible-structure interactions (FFSI) are ubiquitous in engineering and sciences. Peskin’s immersed boundary (IB) method is the first framework for modeling and simulation of such problems. This paper addresses a three-dimensional extension of the IB framework for non-Newtonian fluids which include power-law fluid, Oldroyd-B fluid, and FENE-P fluid. The motion of the non-Newtonian fluids are modelled by the lattice Boltzmann equations (D3Q19 model). The differential constitutive equations of Oldroyd-B and FENE-P fluids are solved by the D3Q7 model. Numerical results indicate that the new method is first-order accurate and conditionally stable. To show the capability of the new method, it is tested on three FFSI toy problems: a power-law fluid past a flexible sheet fixed at its midline, a flexible sheet being flapped periodically at its midline in an Oldroyd-B fluid, and a flexible sheet being rotated at one edge in a FENE-P fluid