Distributed Lagrange Multiplier/fictitious Domain Research Articles

Distributed Lagrange multiplier/fictitious domain finite element method for Stokes/parabolic interface problems with jump coefficients

In this paper, the distributed Lagrange multiplier/fictitious domain (DLM/FD)–mixed finite element method is developed and analyzed for a type of linearized fluid-structure interaction problem–the transient Stokes/parabolic interface problem with jump coefficients. Stability and optimal convergence properties are obtained for both semi- and fully discrete mixed finite element scheme. Numerical experiments are carried out to validate the theoretical results. Similar DLM/FD method can be developed to tackle FSI problems with different compressibility for the fluid and the structure in the future.

Fictitious domain finite element method for Stokes/elliptic interface problems with jump coefficients

In this paper, the distributed Lagrange multiplier/fictitious domain (DLM/FD) finite element method is studied for a generic Stokes/elliptic interface problem with jump coefficients which belongs to a type of linearized stationary fluid–structure interaction problem. A mixed finite element discretization is developed for the proposed DLM/FD method for Stokes/elliptic interface problem and analyzed on the aspects of well-posedness, stability and optimal convergence. Numerical experiments are carried out and the theoretical error estimates of DLM/FD finite element method are validated. • Distributed Lagrange multiplier/fictitious domain finite element method. • Stokes/elliptic interface problem with jump coefficients. • A linearized steady state fluid–structure interaction problem. • New DLM/FD formulation for the incompressible fluid and compressible structure interaction. • Well-posedness, stability and convergence are analyzed for mixed finite elements.

Distributed Lagrange Multiplier/Fictitious Domain Finite Element Method for a Transient Stokes Interface Problem with Jump Coefficients

The distributed Lagrange multiplier/fictitious domain (DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients. The semi- and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface, where the arbitrary Lagrangian-Eulerian (ALE) technique is employed to deal with the moving and immersed subdomain. Stability and optimal convergence properties are obtained for both schemes. Numerical experiments are carried out for different scenarios of jump coefficients, and all theoretical results are validated.

A 3D DLM/FD method for simulating the motion of spheres in a bounded shear flow of Oldroyd-B fluids

We present a novel distributed Lagrange multiplier/fictitious domain (DLM/FD) method for simulating fluid-particle interaction in Oldroyd-B fluids under creeping conditions. The results concerning two ball interaction in a three dimensional (3D) bounded shear flow are obtained for Weissenberg numbers up to 1. The pass and return trajectories of the two ball mass centers are similar to those in a Newtonian fluid; but they lose the symmetry due to the effect of elastic force arising from viscoelastic fluids. A tumbling chain of two balls (a dipole) may occur, depending on the value of the Weissenberg number and the initial vertical displacement of the ball mass center to the middle plane between two walls.

A 3D DLM/FD method for simulating the motion of spheres and ellipsoids under creeping flow conditions

We present in this article a novel distributed Lagrange multiplier/fictitious domain (DLM/FD) method for simulating fluid-particle interaction in three-dimensional (3D) Stokes flow. The methodology is validated by comparing the numerical results for a neutrally buoyant particle, of either spherical or prolate shape, with the associated Jeffrey's solutions for a simple shear flow. The results concerning two balls, interacting under creeping flow conditions in a bounded shear flow, are consistent with those available in the literature. We will discuss also the interactions of two balls in a bounded shear flow, when these balls are very close initially. For a prolate ellipsoid rotating in a shear flow under the sole effect of the particle inertia, shear plane tumbling is stable, while log-rolling is unstable. For two prolate ellipsoids interacting in a bounded shear flow, the results are similar to those for two balls if the major axes are initially orthogonal to the shear plane (a result not at all surprising considering that the intersections of the ellipsoids with the shear pane are circular).

A DLM/FD/IB method for simulating compound vesicle motion under creeping flow condition

In this article we present first a new distributed Lagrange multiplier/fictitious domain (DLM/FD) method for simulating fluid–particle interaction in Stokes flow. A conjugate gradient method driven by both pressure and distributed Lagrange multiplier, called one-shot method, has been developed to solve the discrete Stokes problem while enforcing the rigid body motion within the region occupied by the particle. The methodology is validated by comparing the numerical results of a neutrally buoyant particle of either a circular or elliptic shape with the associated Jeffery's solutions. We have successively combined the above methodology with an immersed boundary (IB) method and an elastic membrane modeled by a spring network to simulate the dynamics of a compound vesicle. In simple shear flow under creeping flow condition, the results are consistent with those obtained in literature. In Poiseuille flow, the compound vesicle motion is dominated by the motion of the vesicle membrane as expected and stays in the central region of the channel.

A fictitious domain method with a hybrid cell model for simulating motion of cells in fluid flow

In this study, we develop a hybrid model to represent membranes of biological cells and use the distributed-Lagrange-multiplier/fictitious-domain (DLM/FD) formulation for simulating the fluid/cell interactions. The hybrid model representing the cellular structure consists of a continuum representation of the lipid bilayer, from which the bending force is calculated through energetic variational approach, a discrete cytoskeleton model utilizing the worm-like chain to represent network filament, and area/volume constraints. For our computational scheme, a formally second-order accurate fractional step scheme is employed to decouple the entire system into three sub-systems: a fluid problem, a solid problem and a Lagrange multiplier problem. The flow problem is solved by the projection method; the solid problem based on the cell model is solved by a combination of level set method, ENO reconstruction, and the Newton method; and the Lagrange multiplier problem is solved by immerse boundary interpolation. The incompressibility of the material is implemented with the penalty function method. Numerical results compare favorably with previously reported numerical and experimental results, and show that our method is suited to the simulation of the cell motion in flow.

이상 유동에서의 유체-구조 연성해석을 위한 Direct Forcing/Ficititious Domain-Level Set Method

In the present paper, a direct forcing/fictitious domain (DF/FD) level set method is proposed to simulate the FSI (fluid-solid interaction) in two-phase flow. The main idea is to combine the direct-forcing/fictitious domain (DF/FD) method with the level set method in the Cartesian coordinates. The DF/FD method is a non-Lagrange-multiplier version of a distributed Lagrange multiplier/fictitious domain (DLM/FD) method. This method does not sacrifice the accuracy and robustness by employing a discrete <TEX>${\delta}$</TEX> (Dirac delta) function to transfer quantities between the Eulerian nodes and Lagrangian points explicitly as the immersed boundary method. The advantages of this approach are the simple concept, easy implementation, and utilization of the original governing equation without modification. Simulations of various water-entry problems have been conducted to validate the capability and accuracy of the present method in solving the FSI in two-phase flow. Consequently, the present results are found to be in good agreement with those of previous studies.

PeliGRIFF, a parallel DEM-DLM/FD direct numerical simulation tool for 3D particulate flows

The problem of particulate flows at moderate to high concentration and finite Reynolds number is addressed by parallel direct numerical simulation. The present contribution is an extension of the work published in Computers & Fluids 38:1608 (2009), where systems of moderate size in a 2D geometry were examined. At the numerical level, the suggested method is inspired by the framework established by Glowinski et al. (Int J Multiph Flow 25:755, 1999) in the sense that their Distributed Lagrange Multiplier/Fictitious Domain (DLM/FD) formulation and their operator-splitting idea are employed. In contrast, particle collisions are handled by an efficient Discrete Element Method (DEM) granular solver, which allows one to consider both smoothly (sphere) and non-smoothly (angular polyhedron) shaped particles. From a computational viewpoint, a basic though efficient strategy has been developed to implement the DLM/FD method in a domain decomposition/distributed fashion. To achieve this goal, the serial code, GRIFF (GRains In Fluid Flow; see Comput Fluids 38:1608–1628, 2009) is upgraded to fully MPI capabilities. The new code, PeliGRIFF (Parallel Efficient Library for GRains in Fluid Flow) is developed under the framework of the fully MPI open-source platform PELICANS. The parallel computing capabilities of PeliGRIFF offer new perspectives in the study of particulate flows and indeed increase the number of particles usually simulated in the literature. Solutions to address new issues raised by the parallelization of the DLM/FD method and assess the scalable properties of the code are proposed. Results on the 2D/3D sedimentation of a significant collection of isometric polygonal/polyhedral particles in a Newtonian fluid with collisions are presented as a validation test and an illustration of the class of particulate flows PeliGRIFF is able to investigate.

Micro-rheology of dense particulate flows: Application to immersed avalanches

Abstract We rely here on a non-smooth contact dynamics (NSCD) approach to treat particle collisions in a direct numerical simulation of a dense particulate flow. Interactions between particles are considered by a non-smooth formulation of particle dynamics at the microscopic scale, which enables one to straightforwardly implement complex contact laws. The hydrodynamic coupling is achieved by a distributed Lagrange multiplier/fictitious domain (DLM/FD) method. As a preliminary step, the relevance of our NSCD-DLM/FD method is assessed by comparing results of 2D sedimentation simulations with those obtained with a usual molecular dynamics collision model. Then, we use it to investigate how a fully immersed granular packing collapses depending on its initial particle volume fraction, providing clues on the micro-rheology of dense particulate flows.

Direct Numerical Simulation of particulate flow with heat transfer

Numerical simulations of heat transfer in non-isothermal particulate flows are important to better understand the flow pattern. The complexity of numerical algorithms coupling the heat and mass transfer and the considerable computational resources required limit the number of such direct simulations that can be reasonably performed. We suggest a Distributed Lagrange Multiplier/Fictitious Domain (DLM/FD) method to compute the temperature distribution and the heat exchange between the fluid and solid phases. The Boussinesq approximation is considered for the flow/temperature fields coupling. We employ a Finite Element Method (FEM) to solve the fluid flow conservation equations for mass, momentum and energy. The motion of particles is computed by a Discrete Element Method (DEM). On each particle, heat transfer is solved using a FEM. For each class of particles, we generate a single FEM grid and translate/rotate it at each time step to match the physical configuration of each particle. Distributed Lagrange multipliers for both the velocity and temperature fields are introduced to treat the fluid/solid interaction. This work is an extension of the method we proposed in Yu et al. (2006). Two two-dimensional (2D) test cases are proposed to validate the implementation by comparing our computational results with those reported in the literature. Finally, the sedimentation of a single sphere in a semi-infinite channel is presented and the results are discussed.

A DEM-DLM/FD method for direct numerical simulation of particulate flows: Sedimentation of polygonal isometric particles in a Newtonian fluid with collisions

An efficient direct numerical simulation method to tackle the problem of particulate flows at moderate to high concentration and finite Reynolds number is presented. Our method is built on the framework established by Glowinski and his co-workers [Glowinski R, Pan TW, Hesla TI, Joseph DD. A distributed lagrange multiplier/fictitious domain method for particulate flow. Int J Multiphase Flow 1999;25:755–94] in the sense that we use their Distributed Lagrange Multiplier/Fictitious Domain (DLM/FD) formulation and their operator-splitting idea but differs in the treatment of particle collisions. Compared to our previous works [Yu Z, Wachs A, Peysson Y. Numerical simulation of particle sedimentation in shear-thinning fluids with a fictitious domain method. J Non Newtonian Fluid Mech 2006;136:126–139; Yu Z, Shao X, Wachs A. A fictitious domain method for particulate flow with heat transfer. J Comput Phys 2006;217:424–52; Yu Z, Wachs A. A fictitious domain method for dynamic simulation of particle sedimentation in Bingham fluids. J Non Newtonian Fluid Mech 2007;145:78–91], the novelty of our present contribution relies on replacing the simple artificial repulsive force based collision model usually employed in the literature by an efficient Discrete Element Method (DEM) granular solver. The use of our DEM solver enables us to consider particles of arbitrary shape (at least convex) and to account for actual contacts, in the sense that particles actually touch each other, in contrast with the repulsive force based collision model. We validate GRIFF, 1 ▪ GRIFF stands for “GRains In Fluid Flow”. 1 our numerical code, against benchmark problems and compare our predictions with those available in the literature. Results, which, to the best of our knowledge, have never been reported elsewhere, on the 2D sedimentation of isometric polygonal particles with collisions are presented.

A fast and robust fictitious domain method for modelling viscous flows in complex mixers: The example of propellant make‐down

AbstractThis work deals with the development of a fast three‐dimensional numerical strategy for the simulation of viscous fluid flow in complex mixing systems. The proposed method is based on a distributed Lagrange multiplier fictitious domain method and the use of the low‐cost MINI finite element. Contrary to the previous fictitious domain method developed by our group a few years ago, the underlying partial differential equations are solved here in a coupled manner using a consistent penalty technique. The method is discussed in detail and its precision is assessed by means of experimental data in the case of an agitated vessel. A comparison made with our existing fictitious domain method and its decoupled Uzawa‐based solver clearly shows the advantages of resorting to the MINI finite element and fully coupled solution strategy. The new technique is then applied to the simulation of the flow of a Newtonian viscous fluid in a three‐blade planetary mixer in the context of the production of solid propellants. Copyright © 2008 John Wiley &amp; Sons, Ltd.

A fictitious domain method for particulate flows with heat transfer

The distributed-Lagrange-multiplier/fictitious-domain (DLM/FD) method of Glowinski et al. [R. Glowinski, T.-W. Pan, T.I. Hesla, D.D. Joseph, A distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. Multiphase Flow 25 (1999) 755–794] is extended to deal with heat transfer in particulate flows in two dimensions. The Boussinesq approximation is employed for the coupling between the flow and temperature fields. The fluid-flow equations are solved with the finite-difference projection method on a half-staggered grid. In our operator splitting scheme, the Lagrange multipliers at the previous time level are kept in the fluid equations, and the new Lagrange multipliers for the rigid-body motion constraint and the Dirichlet temperature boundary condition are determined from the reduced saddle-point problem, whereas a very simple scheme based on the fully explicit computation of the Lagrange multiplier is proposed for the problem in which the solid heat conduction inside the particle boundary is also considered. Our code for the case of fixed temperature on the immersed boundary is verified by comparing favorably our results on the natural convection driven by a hot cylinder eccentrically placed in a square box and on the sedimentation of a cold circular particle in a vertical channel to the data in the literature. The code for the case of freely varying temperature on the boundaries of freely moving particles is applied to analyze the motion of a catalyst particle in a box and in particular the heat conductivities of nanofluids and sheared non-colloidal suspensions, respectively. Our preliminary computational results support the argument that the micro-heat-convection in the fluids is primarily responsible for the unusually high heat conductivity of nanofluids. It is shown that the Peclet number plays a negative role in the diffusion-related heat conductivity of a sheared non-colloidal suspension, whereas the Reynolds number does the opposite.

A DLM/FD method for fluid/flexible-body interactions

In this study, we extended the distributed-Lagrange-multiplier/fictitious-domain (DLM/FD) formulation of Glowinski et al. [Int. J. Multiphase Flow 25 (1999) 755] for the fluid/rigid-body interactions to deal with the fluid/flexible-body interactions by replacing Newton’s equations of motion for the rigid body with the continuum equations for the general solid material. Similar to the rigid-body case where the DLM is introduced as a pseudo body force to enforce the constraint of rigid-body motion of the fictitious fluid in the solid domain, the Lagrange multiplier in our formulation is to enforce the fictitious fluid to move at the same velocity as the solid. For our computational scheme, a first-order accurate fractional step scheme is employed to decouple the entire system into three sub-systems: a fluid problem, a solid problem and a Lagrange multiplier problem; the flow problem is solved with the projection method on half-staggered grids; the solid problem is solved with the Lagrangian finite element method and the Newton iterative method; and the incompressibility of the material is implemented with the penalty function method. The proposed method is applied to two typical fluid–structure interaction problems: the flow-driven oscillation of a flexible plate along the flow direction and the self-sustained oscillation across the flow direction. Both results compare favorably with previously reported numerical and experimental results, and show that our method is suited to the simulation of the motion of an incompressible non-linear elastic material in a fluid.

Viscoelastic mobility problem of a system of particles

In this paper we present a new implementation of the distributed Lagrange multiplier/fictitious domain (DLM) method by making some modifications over the original algorithm for the Newtonian case developed by Glowinski et al. [Int. J. Multiphase Flow 25 (1999) 755], and its extended version for the viscoelastic case by Singh et al. [J. Non-Newtonian Fluid Mech. 91 (2000) 165]. The key modification is to replace a finite-element triangulation for the velocity and a “staggered” (twice coarser) triangulation for the pressure with a rectangular discretization for the velocity and the pressure. The sedimentation of a single circular particle in a Newtonian fluid at different Reynolds numbers, sedimentation of particles in the Oldroyd-B fluid, and lateral migration of a single particle in a Poiseuille flow of a Newtonian fluid are numerically simulated with our code. The results show that the new implementation can give a more accurate prediction of the motion of particles compared to the previous DLM codes and even the boundary-fitted methods in some cases. The centering of a particle and the well-organized Karman vortex street are observed at high Reynolds numbers in our simulation of a particle sedimenting in a Newtonian fluid. Both results obtained using the DLM method and the spectral element method reveal that the direct contribution of the viscoelastic normal stress to the force on a particle in the Oldroyd-B fluid is very important.