Distributed Lagrange Multiplier Method Research Articles

A subcycling/non-subcycling time advancement scheme-based DLM immersed boundary method framework for solving single and multiphase fluid–structure interaction problems on dynamically adaptive grids

In this paper, we present an adaptive implementation of the distributed Lagrange multiplier (DLM) immersed boundary (IB) method on multilevel collocated grids for solving single- and multiphase fluid–structure interaction (FSI) problems. Both a non-subcycling time advancement scheme and a subcycling time advancement scheme, which are applied to time-march the composite grid variables on a level-by-level basis, are presented; these schemes use the same time step size and a different time step size, respectively, on different levels. This is in contrast to the existing adaptive versions of the IB method in the literature, in which coarse- and fine-level variables are simultaneously solved and advanced in a coupled fashion. A force-averaging technique and a series of synchronization operations are constructed to achieve excellent momentum and mass conservation across multiple levels of grid hierarchy. We demonstrate the versatility of the present multilevel framework by simulating problems with various types of kinematic constraints imposed by structures on fluids, such as imposing a prescribed motion, free motion, and time-evolving shape of a solid body. The DLM method is also coupled to a robust level set method-based two-phase fluid solver to simulate challenging multiphase flow problems, including wave energy harvesting using a mechanical oscillator. The capabilities and robustness of the computational framework are validated against a variety of benchmarking single-phase and multiphase FSI problems from the literature, which include a three-dimensional swimming eel model to demonstrate the significant speedup and efficiency that result from employing the present multilevel subcycling FSI scheme.

A coupled distributed Lagrange multiplier (DLM) and discrete element method (DEM) approach to simulate particulate flow with collisions

In this paper, a coupled DLM-DEM numerical approach is proposed to enable particle-particle and particle-wall interactions within the open-source IBAMR (Immersed Boundary Adaptive Mesh Refinement) framework. A soft-sphere DEM collision model is adopted and compared against the particle-particle repulsion force model suggested in the literature. It is demonstrated that the former collision model remains effective at both low and high particle-fluid density ratios, whereas the latter model can lead to unphysical particle-particle interaction dynamics at high density ratios. Moreover, the soft-sphere DEM collision model is based on physical material properties, which is in contrast to an artificial parameter-based repulsion force model. The proposed DLM-DEM approach is validated by simulating the particle-particle collision in drafting, kissing, and tumbling (DKT) phenomenon, and particle-wall collision at various Stokes numbers. Using this approach, we also simulate the interactions between a single colliding particle and a particle cluster and analyze the particles' motion and distribution. Additionally, sedimentation of polygon-shaped particles is also considered to demonstrate the versatility of the proposed approach.

Motion of an inertial squirmer in a density stratified fluid

Abstract

Direct Numerical Simulation of Particles in Spatially Varying Electric Fields †

A numerical scheme is developed to simulate the motion of dielectric particles in the uniform and nonuniform electric fields of microfluidic devices. The motion of particles is simulated using a distributed Lagrange multiplier method (DLM) and the electric force acting on the particles is calculated by integrating the Maxwell stress tensor (MST) over the particle surfaces. One of the key features of the DLM method used is that the fluid-particle system is treated implicitly by using a combined weak formulation, where the forces and moments between the particles and fluid cancel, as they are internal to the combined system. The MST is obtained from the electric potential, which, in turn, is obtained by solving the electrostatic problem. In our numerical scheme, the domain is discretized using a finite element scheme and the Marchuk-Yanenko operator-splitting technique is used to decouple the difficulties associated with the incompressibility constraint, the nonlinear convection term, the rigid-body motion constraint and the electric force term. The numerical code is used to study the motion of particles in a dielectrophoretic cage which can be used to trap and hold particles at its center. If the particles moves away from the center of the cage, a resorting force acts on them towards the center. The MST results show that the ratio of the particle-particle interaction and dielectrophoretic forces decreases with increasing particle size. Therefore, larger particles move primarily under the action of the dielectrophoretic (DEP) force, especially in the high electric field gradient regions. Consequently, when the spacing between the electrodes is comparable to the particle size, instead of collecting on the same electrode by forming chains, they collect at different electrodes.

Application of parallel distributed Lagrange multiplier technique to simulate coupled Fluid-Granular flows in pipes with varying Cross-Sectional area

Fluid-granular flows are common phenomena in nature and industry. Here, an efficient computational technique based on the distributed Lagrange multiplier method is utilized to simulate complex fluid-granular flows. Each particle is explicitly resolved on an Eulerian grid as a separate domain, using solid volume fractions. The fluid equations are solved through the entire computational domain, however, Lagrange multiplier constrains are applied inside the particle domain such that the fluid within any volume associated with a solid particle moves as an incompressible rigid body. The particle–particle interactions are implemented using explicit force-displacement interactions for frictional inelastic particles similar to the DEM method (Cundall and Strack, 1979) with some modifications using the volume of an overlapping region as an input to the contact forces. A parallel implementation of the method is based on the SAMRAI (Structured Adaptive Mesh Refinement Application Infrastructure) library.Application of this method to simulate fluid-granular flows in pipes with a step decrease in cross-sectional area is discussed. Correlations between pressure losses and solid mass fluxes in a pipe for different particle concentrations, pipe diameters and particle sizes are studied. Results of the simulations agree well with available experimental data for the mass flow ratio vs pressure drop during conveying.

Three dimensional fluid–structure interaction under pulsatile flow by using distributed Lagrange multiplier method

In this article we discuss an application of a Lagrange multiplier based fictitious domain method for the simulation of moving leaflets in an unsteady flow generated by pressure gradients during the systolic phase of the cardiac cycle in three dimensional geometry. The mathematical model includes the Navier–Stokes equations coupled with the Euler–Newton equations describing fluid–structure interaction for the generalized Neumann boundary conditions on upstream and downstream boundaries. The solution method includes the finite element method combined with an operator-splitting scheme, where the fictitious formulation allows the flow calculations to be in a fixed rectangular parallelepiped to predict the dynamic structural positions and flow field during the valve opening and closing phases.

Fictitious versus nonfictitious formulation of backward step flow

In this article we discuss the numerical results obtained by two methods to solve the incompressible viscous flow modelled by the Navier–Stokes equations for a backward step flow for Reynold’s number R e = 50 . In the first method we have used a nonfictitious formulation of the problem and compared the result with the alternating direction method obtained by Glowinski. In the second method we have considered the fictitious formulation of the same problem based on the distributed Lagrange multiplier method. Our goal is to observe its behavior at the nonsmooth points, both with matching and nonmatching boundary points. In the fictitious formulation the step is treated as a nonmoving obstacle present in a channel by extending the flow region to a simplified rectangular domain. In this study, we find that our results compare very well for this stationary problem with nonsmooth boundary. Our solution method combines the finite element method and the operator splitting scheme of the Marchuk–Yanenko type.

Electrohydrodynamics of yeast cells in microchannels subjected to travelling electric fields

While travelling wave dielectrophoresis (twDEP) offers a promising method for the control of micro-sized particles suspended in liquids, particularly when the motion of particles along the length of the channel is sought without having to pump the liquid itself, it leads to a variety of complex dynamical regimes which need to be clearly understood for the design of efficient microfluidic devices targeting particular functions. In this paper, we describe the various dynamical regimes in terms of the forces acting on the particles, i.e. the conventional dielectrophoretic and twDEP force and torque, the viscous drag exerted by the fluid on the particle and the electrostatic and hydrodynamic particle–particle interactions. We also explore the variation of the dynamical regimes in three different configurations typical of microfluidic channels whose electrodes are embedded in the bottom wall. The first two configurations have different, i.e. aligned and staggered, electrode geometries, and the third configuration consists of aligned electrodes but energized at different potentials. For these purposes, we use our direct numerical simulation code based on the distributed Lagrange multiplier method for solving the equations of motion for both the fluid and the individual particles, and the point dipole model to compute the electrostatic forces. The model particles are chosen so as to have mechanical and electrical properties of yeast cells suspended in an aqueous solution. It is found that the motion of the particles not only depends significantly on the Clausius–Mossotti factor, which is a function of the electric properties of the fluid and the particles, but also on the specific configuration considered. Particularly, the spinning of particles plays a crucial role in the particle translations and interactions, but the direction of such spinning motion depends on the particular device configuration used.

Manipulation of particles using dielectrophoresis

A numerical scheme based on the distributed Lagrange multiplier method (DLM) is used to study the motion of particles of electrorheological suspensions subjected to non-uniform electric fields. At small Reynolds number, the time taken by the particles to collect at the minimums or maximums of the electric field is primarily determined by a parameter defined to be the ratio of the dielectrophoretic and viscous forces. Simulations show that in non-uniform electric fields the collection time is also influenced by a parameter defined by the ratio of the electrostatic particle–particle interaction and dielectrophoretic forces. The collection time decreases as this parameter decreases because when this parameter is less than one, particles move to the regions of high or low electric field regions individually. However, when this parameter is greater than one, particles regroup into chains which then move toward the electric field maximums or minimums without breaking. It is also shown that when the real part of the Clausius–Mosotti factor ( β) is negative the positions of the local minimums of the electric field, and thus also the locations where particles collect, can be modified by changing the electric potential boundary conditions.

Dielectrophoresis of nanoparticles

A numerical scheme based on the distributed Lagrange multiplier method (DLM) is used to study the motion of nano-sized particles of dielectric suspensions subjected to uniform and nonuniform electric fields. Particles are subjected to both electrostatic and hydrodynamic forces, as well as Brownian motion. The results of the simulations presented in this paper show that uniform electric fields the evolution of the particle structures depends on the ratio of electrostatic particle-particle interactions and Brownian forces. When this ratio is of the order of 100 or greater, particles form stable chains and columns, whereas when it is of the order of 10 or smaller the particle distribution is random. For the nonuniform electric field cases considered in this paper, the relative magnitude of Brownian forces is in the range such that it does not influence the eventual collection of particles by dielectrophoresis and the particular locations where the particles are collected. However, Brownian motion is observed to influence the transient particle trajectories. The deviation of the particle trajectories compared to those determined by the electrostatic and hydrodynamic forces alone is characterized by the ratio of Brownian and dielectrophoretic forces.

Dynamics of Electrorheological Suspensions Subjected to Spatially Nonuniform Electric Fields

A numerical method based on the distributed Lagrange multiplier method (DLM) is developed for the direct simulation of electrorheological (ER) liquids subjected to spatially nonuniform electric field. The flow inside particle boundaries is constrained to be rigid body motion by the distributed Lagrange multiplier method and the electrostatic forces acting on the particles are obtained using the point-dipole approximation. The numerical scheme is verified by performing a convergence study which shows that the results are independent of mesh and time step sizes. The dynamical behavior of ER suspensions subjected to nonuniform electric field depends on the solids fraction, the ratio of the domain size and particle radius, and four additional dimensionless parameters which respectively determine the importance of inertia, viscous, electrostatic particle-particle interaction and dielectrophoretic forces. For inertia less flows a parameter defined by the ratio of the dielectrophoretic and viscous forces, determines the time duration in which the particles collect near either the local maximums or local minimums of the electric field magnitude, depending on the sign of the real part of the Clausius-Mossotti factor. In a channel subjected to a given nonuniform electric field, when the applied pressure gradient is smaller than a critical value, the flow assists in the collection of particles at the electrodes, but when the pressure gradient is above this critical value the particles are swept away by the flow.

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Guest Editorial

Spectral distributed Lagrange multiplier method: algorithm and benchmark tests

We extend the formulation of the distributed Lagrange multiplier (DLM) approach for particulate flows to high-order methods within the spectral/ hp element framework. We implement the rigid-body motion constraint inside the particle via a penalty method. The high-order DLM method demonstrates spectral convergence rate, i.e. discretization errors decrease exponentially as the order of spectral polynomials increases. We provide detailed comparisons between the spectral DLM method, direct numerical simulations, and the force coupling method for a number of 2D and 3D benchmark flow problems. We also validate the spectral DLM method with available experimental data for a transient problem. The new DLM method can potentially be very effective in many-moving body problems, where a smaller number of grid points is required in comparison with low-order methods.

Distributed Lagrange multiplier method for particulate flows with collisions

A modified distributed Lagrange multiplier/fictitious domain method (DLM) that allows particles to undergo collisions is developed for particulate flows. In the earlier versions of the DLM method for Newtonian and viscoelastic liquids the particle surfaces were restricted to be more than one velocity element away from each other. A repulsive body force was applied to the particles when the distance between them was smaller than this critical value. This was necessary for ensuring that conflicting rigid body motion constraints from two different particles are not imposed at the same velocity nodes. In the modified DLM method the particles are allowed to come arbitrarily close to each other and even slightly overlap each other. When conflicting rigid body motion constraints from two different particles are applicable on a velocity node, the constraint from the particle that is closer to that node is used and the other constraint is dropped. An elastic repulsive force is applied when the particles overlap each other. In our simulations, the particles are allowed to overlap as much as one hundredth of the velocity element size. The modified DLM method is implemented for both Newtonian and viscoelastic liquids. Our simulations show that when particles are dropped in a channel, and the viscoelastic Mach number ( M) is less than one and the elasticity number ( E) is greater than one, the particles form a chain parallel to the flow direction. As in experiments, the new method allows particles in the chain to approximately touch each other. The particles dropped in a Newtonian liquid, on the other hand, undergo characteristic drafting, kissing and tumbling. During the touching phase, as in experiments, the two particles touch each other. The modified method thus allows hydrodynamic forces to be fully resolved to within the tolerance of the mesh and thus the extra artificial force in a security zone outside the particle which are used in all other methods are not needed.