Implicit Immersed Boundary Method Research Articles

An implicit immersed boundary method for Robin boundary condition

In this work, a novel boundary condition-enforced immersed boundary method (IBM) for simulating complex thermal–fluid–structure interaction (TFSI) problems with Robin boundary conditions is proposed. In this work, two auxiliary layers of Lagrangian points are introduced and placed within the inner and outer parts of the immersed object to enable the simultaneous evaluation of the temperature and its gradient on the solid surface. The mutual thermal interactions between the immersed object and the surrounding flow are taken into account through the temperature corrections on the Lagrangian points by accurately enforcing the Robin boundary condition on the solid boundary. Hence, a system of linear equations can be formulated to compute the temperature corrections. Subsequently, the temperature corrections are biasedly distributed to the Eulerian points located in the inner auxiliary layer to eliminate the diffusion generated by the smooth delta function on the temperature field outside the solid object. The proposed IBM integrated with the reconstructed thermal lattice Boltzmann flux solver (RTLBFS) is extensively and critically assessed with classical benchmark cases ranging from two-dimensional stationary problems to three-dimensional problems involving moving boundaries to demonstrate the robustness and accuracy of the proposed method for Robin boundary conditions. Results from the present work are extensively corroborated with previous works that adopt different approaches for imposing the Robin boundary condition. It shows that the proposed method not only provides a much more accurate representation of the Robin boundary condition but also introduces a simple implementation for the complex Robin boundary condition within the diffuse interface IBM framework.

Numerical modeling of fluid–structure–piezoelectric interaction for energy harvesting

In the present work, we propose a numerical coupling scheme combining the corotational Finite Element Method (FEM) for solid beams, the regularized Lattice Boltzmann Method (LBM) for fluid flows and a PiezoElectric Model (PEM) for energy harvesting module. The proposed coupling method can be employed to numerically solve Fluid–Structure–PiezoElectric Interaction (FSPEI) multiphysics problems which are the fundamental phenomena in the considered energy harvesting applications. In the proposed coupling method, the FEM and LBM are strongly coupled through the implicit Immersed Boundary Method (IBM) ensuring exactly the no-slip condition. Comparing with volumetric FEMs, the adopted corotational beam formulation can not only allow for large structural displacements, but also help improve the computational efficiency, as the solid structures for energy harvesting devices are usually slender under large motion. The piezoelectric effects are incorporated into the corotational beam dynamics with the help of the virtual work principle. The FEM, LBM and PEM are coupled together in a strong way in the sense that the time integrations are carried out without staggered communications. The proposed coupling scheme is assessed by a series of validation test-cases with increasing complexity, in which a good agreement is obtained with references.

Numerical Study of Flat Plate Impact on Water Using a Compressible CIP–IBM–Based Model

Due to the entrapment of compressible air, the process of a flat plate impact on water is complicated, which cannot be reproduced using incompressible simulations. To investigate such a slamming process, an accuracy compressible fluid–structure interaction numerical model has been proposed. The solution of this model is based on the constrained interpolation profile (CIP) method to solve the Navier–Stokes equations for the computation of fluid, and an implicit immersed boundary method (IBM) is used to calculate the fluid–structure interaction. Firstly, the present (CIP–IBM–based) model is validated against the problem of the flow past a stationary cylinder. Then it is implemented to simulate the problem of a rigid flat plate impact on water. The predicated impact pressure is compared with reference experiments and other simulations. The CIP–IBM–based model shows a better performance in dealing with sub-atmospheric pressure and reloading. From the numerical view, it is shown that the oscillation of the slamming pressure is significantly affected by the compression and expansion of the entrapped air cushion, and under the influence of the air cushion, the slamming pressure distribution along the bottom is not constant, which also varies greatly with time.

A partitioned framework for coupling LBM and FEM through an implicit IBM allowing non-conforming time-steps: Application to fluid-structure interaction in biomechanics

This paper presents a partitioned framework for the numerical simulation of fluid-structure interactions by coupling the lattice Boltzmann method (LBM) and the finite element method (FEM). The two numerical methods LBM and FEM are coupled with an implicit immersed boundary method (IBM) in a strong way, which ensures exactly the no-slip condition and the continuities of velocity and stress at the fluid-solid interface and each instant in time. In the proposed partitioned coupling procedure, the coupling system of equations are first established and then condensed to the interface. By solving the condensed coupling system of equations, the interface force field is obtained and sent to both solvers to accomplish time integrations in each sub-domain. In addition, two strategies based on linear interpolation in time are proposed to handle the cases with non-conforming time-steps in the fluid and solid sub-domains. Through several 2D and 3D numerical test-cases on the mechanical heart valve, the fluid-induced vibration of a deformable solid beam, the flapping flag, the proposed coupling framework is validated with good agreements with references. Finally, a test-case on the interaction between the blood flow and the aortic valve is carried out, showing the applicability of the present framework in realistic biomechanical applications.

Effects of non-sinusoidal pitching motion on energy input performance of micro-linear turbine cascade

In this paper, a numerical simulation of a micro-linear turbine cascade is presented. The upstream stay vane and downstream guide vane were surrounded by incompressible flow, and the guide vane was modeled with forced pitching motions. The implicit immersed boundary method was used as the moving boundary flow solver. To seek a better understanding of the influence of the motion parameters on hydrodynamic characteristics and input power of the oscillating guide vane, the forced motion of a non-sinusoidal oscillating guide vane was first investigated for different reduced frequencies and profile parameters. Second, the instantaneous hydrodynamic behavior was simulated for different motion combinations and Reynolds numbers. The results showed that the reduced frequencies and profile parameters of the pitch movement had a large influence on the input power, and the peak of the input power at f∗ = 0.22 was nearly 9.41 times that at f∗ = 0.074, but the input power was not sensitive to the motion combination and Reynolds number. However, the Reynolds number had a large influence on the drag, and the maximum drag coefficient for Re = 6.65 × 105 was 4.566; however, it was only 0.5912 for Re = 6.65 × 104.

On the implicit immersed boundary method in coupled discrete element and lattice Boltzmann method

SummaryThe coupled discrete element method and lattice Boltzmann method (DEMLBM) has increasingly drawn attention of researchers in geomechanics due to its mesoscopic nature since 2000. Immersed boundary method (IBM) and immersed moving boundary (IMB) are two popular schemes for coupling fluid particle in DEMLBM. This work aims at coupling DEM and LBM using the latest IBM algorithm and investigating its accuracy, computational efficiency, and applicability. Two benchmark tests, interstitial fluid flow in an ideal packing and single particle sedimentation in viscous fluid, are carried out to demonstrate the accuracy of IBM through semi‐empirical Ergun equation, finite element method (FEM), and IMB. Then, simulations of particle migration with relatively large velocity in Poiseuille flow are utilized to address limitations of IBM in DEMLBM modeling. In addition, advantages and deficiencies of IBM are discussed and compared with IMB. It is found that the accuracy of IBM can be only guaranteed when sufficient boundary points are used and it is not suitable for geomechanical problems involving large fluid or particle velocity.

A generalized minimal residual method-based immersed boundary-lattice Boltzmann flux solver coupled with finite element method for non-linear fluid-structure interaction problems

A generalized minimal residual method (GMRES) based immersed boundary-lattice Boltzmann flux solver (IB-LBFS) coupled with the finite element method (FEM) is presented in this paper for nonlinear fluid-structure interaction (FSI) problems. This approach effectively combines LBFS for the simulation of the flow field, the total Lagrangian FEM for the evaluation of nonlinear structural deformations, and the immersed boundary method (IBM) for the exchange of information on the fluid-solid interface and implementation of boundary conditions. Both the multidirect forcing and the implicit IBM are considered to examine their effects on numerical accuracy and efficiency. Through numerical simulations on flow past a cylinder, it is shown that the implicit IBM with the GMRES for the linear equation system is more efficient and accurate, which justify the conventional misunderstanding that implicit IBM is always less efficient than explicit methods. Numerical simulations on the lid-driven cavity flow in an inclined cavity, incompressible flows of a uniformly accelerated vertical plate, and the flow induced vibrations of a beam attached behind a cylinder in a channel are also successfully carried out and the obtained results are in good agreement with the published data, which verify the reliability and flexibility of the proposed solver for simulating nonlinear FSI problems. After that, the external flows past two hyperelastic cylinder-beam structures at the Reynolds number of 40–300 are studied and three different modes of static, linear, and nonlinear deformations of the beam are obtained, demonstrating its capability of simulating flows with nonlinear FSI problems with multiple deformable objects.

An IB-LBM implementation for fluid-solid interactions with an MLS approximation for implicit coupling

• Extend the implicit velocity correction method to simulate complex fluid solid interaction. • Implicit immersed boundary method to combined IMLS is first presented. • IMLS approximation with the weight orthogonal basis functions for stable solution. This paper presents a combination of the implicit immersed boundary (IB) method and the lattice Boltzmann method (LBM) for a fluid-solid interaction involving an immersed body with moving boundaries and complex geometries. In the implicit IB-LBM method, the flow is computed using LBM. The body force caused by the immersed body is not pre-calculated, but implicitly determined using a corrected velocity field that accurately satisfies the no-slip and no-penetration conditions on the interface between the fluid and solid. The information transfer is not using the traditional Dirac delta function and moving-least-square (MLS). But an improved MLS (IMLS) approximation has been developed to combined the implicit IB-LBM method, where the orthogonal function system with a weight function is used as the basis function and overcome these disadvantages of traditional MLS, in which it may cause numerical instabilities because the matrix inversion must be solved and the final equations system is easily ill-conditioned or singular. Several different flow problems (a two-dimensional flow past a static, rotating and oscillating cylinder, and the vortex-induced vibration of a cylinder as well as the free movement of flapping foil) are simulated to validate the accuracy and efficiency of the present implicit IB-LBM method. The simulation results show good agreement with previous numerical and experimental results. It is found that the present IB-LBM is efficient and reliable in dealing with fluid–solid interaction problems.

Numerical simulation of h-adaptive immersed boundary method for freely falling disks

In this work, a freely falling disk with aspect ratio 1/10 is directly simulated by using an adaptive numerical model implemented on a parallel computation framework JASMIN. The adaptive numerical model is a combination of the h-adaptive mesh refinement technique and the implicit immersed boundary method (IBM). Our numerical results agree well with the experimental results in all of the six degrees of freedom of the disk. Furthermore, very similar vortex structures observed in the experiment were also obtained.

Moving immersed boundary method

SummaryA novel implicit immersed boundary method of high accuracy and efficiency is presented for the simulation of incompressible viscous flow over complex stationary or moving solid boundaries. A boundary force is often introduced in many immersed boundary methods to mimic the presence of solid boundary, such that the overall simulation can be performed on a simple Cartesian grid. The current method inherits this idea and considers the boundary force as a Lagrange multiplier to enforce the no‐slip constraint at the solid boundary, instead of applying constitutional relations for rigid bodies. Hence excessive constraint on the time step is circumvented, and the time step only depends on the discretization of fluid Navier‐Stokes equations, like the CFL condition in present work. To determine the boundary force, an additional moving force equation is derived. The dimension of this derived system is proportional to the number of Lagrangian points describing the solid boundaries, which makes the method very suitable for moving boundary problems since the time for matrix update and system solving is not significant. The force coefficient matrix is made symmetric and positive definite so that the conjugate gradient method can solve the system quickly. The proposed immersed boundary method is incorporated into the fluid solver with a second‐order accurate projection method as a plug‐in. The overall scheme is handled under an efficient fractional step framework, namely, prediction, forcing, and projection. Various simulations are performed to validate current method, and the results compare well with previous experimental and numerical studies.

A coupled Volume of Fluid and Immersed Boundary Method for simulating 3D multiphase flows with contact line dynamics in complex geometries

A numerical methodology is presented for simulating 3D multiphase flows through complex geometries on a non-body conformal Cartesian computational grid. A direct forcing implicit immersed boundary method (IBM) is used to sharply resolve complex geometries, employing the finite volume method (FVM) on a staggered grid. The fluid-fluid interface is tracked by a mass conservative sharp interface volume of fluid (VOF) method. Contact line dynamics at macroscopic length scale is simulated by imposing the apparent contact angle (static or dynamic) as a boundary condition at the three-phase contact line. The developed numerical methodology is validated for several test cases including the equilibrium shape of a droplet on flat and spherical surfaces, the temporal evolution of a droplet spreading on a flat surface. The obtained results show an excellent correspondence with those derived analytically or taken from literature. Furthermore, the present model is used to estimate, on a pore-scale, the residual oil remaining in idealized porous structures after water flooding, similar to the process used in enhanced oil recovery (EOR).

Implicit immersed boundary method for fluid-structure interaction

A novel immersed boundary method is introduced for simulating the fluid-structure interaction problem. Unlike the body-conforming mesh method which imposes the no-slip boundary conditions directly on the immersed interface, the immersed boundary method adopts a boundary force for the presence of the immersed solid. Therefore, the fluid is simply simulated on a fixed Cartesian mesh irrespective the movement of the immersed solid, which circumvents the mesh quality issue that frequently happens in the body-conforming mesh method. To enforce the correct boundary condition, we derive an additional moving force equation. This additional equation is integrated into a second order accurate fractional step method and solved with the conjugate gradient method. The proposed method is validated with several one-way fluid-structure interaction examples.

An immersed boundary-lattice boltzmann flux solver in a moving frame to study three-dimensional freely falling rigid bodies

An immersed boundary-lattice Boltzmann flux solver (IB-LBFS) in a moving frame is proposed in this paper to study three-dimensional (3D) freely falling rigid bodies in unbounded domains. This solver includes the predictor and corrector steps. In the predictor step, the intermediate flow field is predicted on a moving Cartesian grid by 3D LBFS. In the corrector step, velocity correction is made by the implicit immersed boundary method to accurately satisfy the no-slip boundary condition. The motion of rigid body with six degrees of freedom is obtained by solving the governing ordinary differential equations. By enforcing the speed of the Cartesian grid the same as the translational velocity of the rigid body, the present solver is able to study a freely falling object in a large flow domain. It not only extends the applicability of fixed grid-based LBFS but also considerably reduces the number of grid points and computational efforts. In addition, the re-meshing process, which is commonly used in the conventional arbitrary-Lagrangian-Eulerian (ALE) approaches, is avoided. Several benchmarks, including an actively moving sphere, sedimentation of a particle and freely falling disks with finite aspect ratios, are studied to examine the reliability of the proposed solver. The obtained results compare well with theoretical and/or experimental data. After that, a numerical study of an infinitely thin disk falling in four different modes is carried out, which successfully demonstrate the capability of the proposed solver in capturing complex modes of a falling body.

A simple and efficient implicit direct forcing immersed boundary model for simulations of complex flow

This paper presents an implementation of an implicit immersed boundary (IB) method in a flow solver based on the fractional step method and the finite volume method for complex flows involving moving boundaries and complex geometries. In this implementation, a body force caused by the immersed body is first introduced into the N-S equation to model the effect of immersed boundary. However, the body force is not pre-calculated, but implicitly determined in such a way that the velocity at the immersed boundary interpolated from the corrected velocity field accurately satisfies the no-slip and no-penetration conditions. Then, the large-eddy simulation is applied in the solver, where the subgrid-scale stress is determined by the Smagorinsky–Lilly model. Near the immersed boundaries, the subgrid-scale stress is determined by a wall model where the wall shear stress is directly calculated from the Lagrangian force(which represents the action of fluid on solid) on the immersed boundary. Such treatment makes the simulations of high Reynolds number turbulent flows feasible with the IB method. The accuracy and capability of the present method are demonstrated by simulations of a variety of both two- and three-dimensional simulations, including laminar flow past static and oscillating cylinders, rotating hydrofoil and turbulent flow around a three-dimensional circular cylinder and a sphere. It shows that the present implementation provides an easy-to-use, inexpensive and accurate technique for computational fluid dynamics in industrially relevant problems.

Direct numerical simulation of effective drag in dense gas–liquid–solid three-phase flows

Gas–liquid–solid three phase flows are commonly found in (bio-)chemical processes, e.g. in bubble slurry columns in the Fischer–Tropsch process. In order to facilitate the rational scale-up and the design of such columns, a detailed understanding of the complex phase interactions is required. In this work, we focus on the effective drag acting on particles and bubbles, using Direct Numerical Simulations. We combined the Front Tracking method of Roghair et al. (2013b) and the second order implicit Immersed Boundary method of Deen et al. (2012). The resulting hybrid Front Tracking Immersed Boundary method is able to simulate dense three phase flows and is used to study swarm effects in terms of drag. A correlation has been obtained for the drag coefficient for a system consisting of 2mm bubbles and 1mm particles at several phase volume fractions. In future research, the developed methodology can be applied to study the effect of the bubble and particle size and their ratio.

Immersed boundary method (IBM) based direct numerical simulation of open‐cell solid foams: Hydrodynamics

A sharp interface implicit immersed boundary method is developed and used for direct numerical simulations of the flow through open‐cell solid foams with a cellular structure. The complex solid structure of the foam is resolved on a non‐boundary fitted Cartesian computational‐grid. A single representative unit cell of the foam is considered in a periodic domain, and its geometry is approximated based on the structural packing of a tetrakaidecahedron. Simulations are performed for a wide range of porosities (0.638–0.962) and Reynolds numbers (0–500). Flow is enforced by applying a constant body force (momentum source) for three different flow directions along the {100}, {110}, and {111} lattice‐vectors. The drag force on the foam is calculated and a non‐dimensional drag/pressure drop correlation is proposed that fits the entire data set with an average deviation of 5.6%. Moreover, the accurate numerical simulations have helped to elucidate the detailed fluid‐solid interaction in complex porous media. © 2016 The Authors AIChE Journal published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers AIChE J, 63: 1152–1173, 2017

A Coupling Model of DEM and LBM for Fluid Flow through Porous Media

A two-dimensional coupling model of the discrete element method (DEM) and the lattice Boltzmann method (LBM) was used to study the fluid flow through porous media. The fluid flow was modeled at a mesoscale level by using the LBM. The particle motion was modeled at microscale level by means of the DEM. The interfacial interaction between the fluid and the solids is resolved by an implicit immersed boundary method (IBM). The proposed method has validated by comparing the numerical results with theoretic solutions in the literature. It demonstrates the ability of the proposed method to simulate fluid flows through complex porous media.

Deformation of a single red blood cell in a microvessel

Red blood cells (RBCs) are the most common type of cells in human blood and they exhibit different types of motions and deformed shapes in capillary flows. The behaviour of the RBCs should be studied in order to explain the RBC motion and deformation mechanism. This article presents a numerical simulation method for RBC deformation in microvessels. A two dimensional spring network model is used to represent the RBC membrane, where the elastic stretch/compression energy and the bending energy are considered with the constraint of constant RBC surface area. The forces acting on the RBC membrane are obtained from the principle of virtual work. The whole fluid domain is discretized into a finite number of particles using smoothed particle hydrodynamics concepts and the motions of all the particles are solved using Navier--Stokes equations. Minimum energy concepts are used to simulate the deformed shape of the RBC model. To verify the model, the motion of a single RBC is simulated in a Poiseuille flow and the characteristic parachute shape of the RBC is observed. Further simulations reveal that the RBC shows a tank treading motion when it flows in a linear shear flow. References D. A. Fedosov, B. Caswell, and G. E. Karniadakis. A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophys. J. , 98(10):2215–2225, 2010. doi:10.1016/j.bpj.2010.02.002 T. M. Fischer, M. Stohr-Lissen, and H. Schmid-Schonbein. The red cell as a fluid droplet: tank tread-like motion of the human erythrocyte membrane in shear flow. Science , 202(4370):894–896, 1978. doi:10.1126/science.715448 R. A. Frcitas. Exploratory design in medical nanotechnology: a mechanical artificial red cell. Artif. Cell. Blood. Sub. , 26(4):411–430, 1998. doi:10.3109/10731199809117682 H. N. P. Gallage, Y. T. Gu, S. C. Saha, W. Senadeera, and A. Oloyede. Numerical simulation of red blood cells' deformation using SPH method. In Y. T. Gu and S. C. Saha, editors, 4th International Conference on Computational Methods (ICCM 2012) , Crowne Plaza, Gold Coast, QLD, November 2012. H. N. P. Gallage, Y. T. Gu, S. C. Saha, W. Senadeera, and A. Oloyede. Numerical simulation of red blood cells' motion : a review. In Y. T. Gu and S. C. Saha, editors, 4th International Conference on Computational Methods (ICCM 2012) , Crowne Plaza, Gold Coast, QLD, November 2012. Y. T. Gu. Meshfree methods and their comparisons. Int. J. Comput. Meth. , 2(04):477–515, 2005. doi:10.1142/S0219876205000673 D. V. Le, J. White, J. Peraire, K. M. Lim, and B. C. Khoo. An implicit immersed boundary method for three-dimensional fluid–membrane interactions. J. Comput. Phys. , 228(22):8427–8445, 2009. doi:10.1016/j.jcp.2009.08.018 G. R. Liu and Y. T. Gu. An introduction to meshfree methods and their programming . Springer, 2005. G. R. Liu and M. B. Liu. Smoothed particle hydrodynamics: a meshfree particle method . World Scientific, 2003. doi:10.1142/5340 T. W. Pan and T. Wang. Dynamical simulation of red blood cell rheology in microvessels. Int. J. Numer. Anal. Mod. , 6:455–473, 2009. L. Shi, T. W. Pan, and R. Glowinski. Deformation of a single red blood cell in bounded Poiseuille flows. Phys. Rev. E , 85(1):016307, 2012. doi:10.1103/PhysRevE.85.016307 C. Sun and L. L. Munn. Particulate nature of blood determines macroscopic rheology: a 2-D lattice Boltzmann analysis. Biophys. J. , 88(3):1635–1645, 2005. doi:10.1529/biophysj.104.051151 K. I. Tsubota, S. Wada, and T. Yamaguchi. Particle method for computer simulation of red blood cell motion in blood flow. Comput. Meth. Prog. Bio. , 83(2):139–146, 2006. doi:10.1016/j.cmpb.2006.06.005 K. I. Tsubota, S. Wada, and T. Yamaguchi. Simulation study on effects of hematocrit on blood flow properties using particle method. J. Biomech. Sci. Eng. , 1(1):159–170, 2006. doi:10.1299/jbse.1.159 A. Vadapalli, D. Goldman, and A. S. Popel. Calculations of oxygen transport by red blood cells and hemoglobin solutions in capillaries. Artif. Cell. Blood. Sub. , 30(3):157–188, 2002. doi:10.1081/BIO-120004338

A front-tracking method with Catmull–Clark subdivision surfaces for studying liquid capsules enclosed by thin shells in shear flow

This paper presents a front-tracking method for studying the large deformation of a liquid capsule enclosed by a thin shell in a shear flow. The interaction between the fluid and the shell body is accomplished through an implicit immersed boundary method. An improved thin-shell model for computing the forces acting on the shell middle surface during the deformation is described in surface curvilinear coordinates and within the framework of the principle of virtual displacements. This thin-shell model takes full account of in-plane tensions and bending moments developing due to the shell thickness and a preferred three-dimensional membrane structure. The approximation of the shell middle surface is performed through the use of the Catmull–Clark subdivision surfaces. The resulting limit surface is C 2-continuous everywhere except at a small number of extraordinary nodes where it retains C 1 continuity. The smoothness of the limit surface significantly improves the ability of our method in simulating capsules enclosed by hyperelastic thin shells with different shapes and physical properties. The present numerical technique has been validated by several examples including an inflation of a spherical shell and deformations of spherical, ellipsoidal and biconcave capsules in the shear flow. In addition, different types of motion such as tank-treading, swinging, tumbling and transition from tumbling to swinging have been studied over a range of shear rates, viscosity ratios and bending modulus.

A three dimensional implicit immersed boundary method with application

Most algorithms of the immersed boundary method originated by Peskin are explicit when it comes to the computation of the elastic forces exerted by the immersed boundary to the fluid. A drawback of such an explicit approach is a severe restriction on the time step size for maintaining numerical stability. An implicit immersed boundary method in two dimensions using the lattice Boltzmann approach has been proposed. This paper reports an extension of the method to three dimensions and its application to simulation of a massive flexible sheet interacting with an incompressible viscous flow.