Fictitious Domain Method Research Articles

A one-field monolithic fictitious domain method for fluid–structure interactions

In this article, we present a one-field monolithic fictitious domain (FD) method for simulation of general fluid–structure interactions (FSI). “One-field” means only one velocity field is solved in the whole domain, based upon the use of an appropriate L2 projection. “Monolithic” means the fluid and solid equations are solved synchronously (rather than sequentially). We argue that the proposed method has the same generality and robustness as FD methods with distributed Lagrange multiplier (DLM) but is significantly more computationally efficient (because of one-field) whilst being very straightforward to implement. The method is described in detail, followed by the presentation of multiple computational examples in order to validate it across a wide range of fluid and solid parameters and interactions.

Fictitious domain method with boundary value correction using penalty-free Nitsche method

AbstractIn this paper, we consider a fictitious domain approach based on a Nitsche type method without penalty. To allow for high order approximation using piecewise affine approximation of the geometry we use a boundary value correction technique based on Taylor expansion from the approximate to the physical boundary. To ensure stability of the method a ghost penalty stabilization is considered in the boundary zone. We prove optimal error estimates in the

Hydrodynamics characterization of a choanoid fluidized bed bioreactor used in the bioartificial liver system: Fully resolved simulation with a fictitious domain method

Choanoid fluidized bed bioreactors (CFBBs) are newly developed core devices used in bioartificial liver-support systems to detoxify blood plasma of patients with microencapsulated liver cells. Direct numerical simulations (DNS) with a direct-forcing/fictitious domain (DF/FD) method were conducted to study the hydrodynamic performance of a CFBB. The effects of particle–fluid density ratio, particle number, and filter screens preventing particles flowing out of the reactor were investigated. Depending on density ratio, two flow patterns are evident: the circulation mode in which the suspension rises along one sidewall and descends along the other sidewall, and the non-circulation mode in which the whole suspension roughly flows upward. The circulation mode takes place under non-neutral-buoyancy where the particle sedimentation dominates, whereas the non-circulation mode occurs under pure or near-neutral buoyancy with particle–fluid density ratios of unity or near unity. With particle–fluid density ratio of 1.01, the bioartificial liver reactor performs optimally as the significant particle accumulation existing in the non-circulation mode and the large shear forces on particles in the circulation mode are avoided. At higher particle volume fractions, more particles accumulate at the filter screens and a secondary counter circulation to the primary flow is observed at the top of the bed. Modelled as porous media, the filter screens play a negative role on particle fluidization velocities; without screens, particles are fluidized faster because of the higher fluid velocities in the jet center region. This work extends the DF/FD-based DNS to a fluidized bed and accounts for effects from inclined side walls and porous media, providing some hydrodynamics insight that is important for CFBB design and operation optimization.

The fictitious domain method with penalty for the parabolic problem in moving-boundary domain: The error estimate of penalty and the finite element approximation

We consider the fictitious domain method with penalty for the parabolic problem in a moving-boundary domain. Two types of penalty (the H1 and L2-penalty methods) are investigated, for which we obtain the error estimate of penalty. Moreover, for H1-penalty method, the H2-regularity and a-priori estimate depending on the penalty parameter ε are obtained. We apply the finite element method to the H1-penalty problem, and obtain the stability and error estimate for the numerical solution. The theoretical results are confirmed by the numerical experiments.

A novel CFS‐PML boundary condition for transient electromagnetic simulation using a fictitious wave domain method

AbstractIn this study, we apply fictitious wave domain (FWD) methods, based on the correspondence principle for the wave and diffusion fields, to finite difference (FD) modeling of transient electromagnetic (TEM) diffusion problems for geophysical applications. A novel complex frequency shifted perfectly matched layer (PML) boundary condition is adapted to the FWD to truncate the computational domain, with the maximum electromagnetic wave propagation velocity in the FWD used to set the absorbing parameters for the boundary layers. Using domains of varying spatial extent we demonstrate that these boundary conditions offer significant improvements over simpler PML approaches, which can result in spurious reflections and large errors in the FWD solutions, especially for low frequencies and late times. In our development, resistive air layers are directly included in the FWD, allowing simulation of TEM responses in the presence of topography, as is commonly encountered in geophysical applications. We compare responses obtained by our new FD‐FWD approach and with the spectral Lanczos decomposition method on 3‐D resistivity models of varying complexity. The comparisons demonstrate that our absorbing boundary condition in FWD for the TEM diffusion problems works well even in complex high‐contrast conductivity models.

The Technique of the Immersed Boundary Method: Application to Solving Shape Optimization Problem

We present a numerical method based on genetic algorithm combined with a fictitious domain method for a shape optimization problem governed by an elliptic equation with Dirichlet boundary condition. The technique of the immersed boundary method is incorporated into the framework of the fictitious domain method for solving the state equations. Contrary to the conventional methods, our method does not make use of the finite element discretization with obstacle fitted meshes. It conduces to overcoming difficulties arising from re-meshing operations in the optimization process. The method can lead to a reduction in computational effort and is easily programmable. It is applied to a shape reconstruction problem in the airfoil design. Numerical experiments demonstrate the efficiency of the proposed approach.

Fictitious Domain Method Combined with the DEM for Studying Particle-Particle/Particle-Wall Collision in Fluid

The interaction between particles and a fluid is very important in many fields. The hydrodynamic characteristics of both spheroidal particles and non-spherical particles sediment in the fluid and their rebound dynamics were investigated by using the fictitious domain method-discrete element method (FDM-DEM) in this paper. The novelty lies in the combination of the Monte Carlo scheme with FDM to improve the accuracy of hydrodynamic force. A soft-sphere scheme of DEM is used to model the collision of particles. The hydrodynamic force on the particle is fully resolved by the FDM. The numerical results of spherical particle are verified by comparing the previous numerical and experimental results implemented by Cate. The results of non-spherical particles sediment show that the path instability occurs. Those results are in good agreement with the corresponding published data. The method presented in this paper can be used in practical applications.

Numerical simulation of 2D unsteady shear-thinning non-Newtonian incompressible fluid in screw extruder with fictitious domain method

In this paper, we develop a fictitious domain method with Distributed Lagrange Multipliers for simulating 2D unsteady shear-thinning non-newtonian incompressible flow in a single-screw and twin-screw extruder. The advantage of the fictitious domain method is that one can use a fixed mesh even as the fluid domain changes in time (as the screws are rotating), eliminating the need for repeated remeshing and projection. The method uses a finite element discretization in space and an operator-splitting technique for discretization in time. Numerical results are given for the flow inside a single-screw extruder or twin-screw extruder, which show that the fictitious domain method is efficient for the problem.

A fictitious domain method for frictionless contact problems in elasticity using Nitsche’s method

In this paper, we develop and analyze a finite element fictitious domain approach based on Nitsche's method for the approximation of frictionless contact problems of two deformable elastic bodies. In the proposed method, the geometry of the bodies and the boundary conditions, including the contact condition between the two bodies, are described independently of the mesh of the fictitious domain. We prove that the optimal convergence is preserved. Numerical experiments are provided which confirm the correct behavior of the proposed method.

A Fictitious Domain Method with Distributed Lagrange Multiplier for Parabolic Problems With Moving Interfaces

In this paper, we study the fictitious domain method with distributed Lagrange multiplier for the jump-coefficient parabolic problems with moving interfaces. The equivalence between the fictitious domain weak form and the standard weak form of a parabolic interface problem is proved, and the uniform well-posedness of the full discretization of fictitious domain finite element method with distributed Lagrange multiplier is demonstrated. We further analyze the convergence properties for the fully discrete finite element approximation in the norms of $$L^2$$ , $$H^1$$ and a new energy norm. On the other hand, we introduce a subgrid integration technique in order to allow the fictitious domain finite element method to be performed on the triangular meshes without doing any interpolation between the authentic domain and the fictitious domain. Numerical experiments confirm the theoretical results, and show the good performances of the proposed schemes.

Effect of confinement in wall-bounded non-colloidal suspensions

This paper presents three-dimensional numerical simulations of non-colloidal dense suspensions in a wall-bounded shear flow at zero Reynolds number. Simulations rely on a fictitious domain method with a detailed modelling of particle–particle and wall–particle lubrication forces, as well as contact forces including particle roughness and friction. This study emphasizes the effect of walls on the structure, velocity and rheology of a moderately confined suspension (channel gap to particle radius ratio of 20) for a volume fraction range $0.1\leqslant {\it\phi}\leqslant 0.5$. The wall region shows particle layers with a hexagonal structure. The size of this layered zone depends on volume fraction and is only weakly affected by friction. This structure implies a wall slip which is in good accordance with empirical models. Simulations show that this wall slip can be mitigated by reducing particle roughness. For ${\it\phi}\lessapprox 0.4$, wall-induced layering has a moderate impact on the viscosity and second normal stress difference $N_{2}$. Conversely, it significantly alters the first normal stress difference $N_{1}$ and can result in positive $N_{1}$, in better agreement with some experiments. Friction enhances this effect, which is shown to be due to a substantial decrease in the contact normal stress $|{\it\Sigma}_{xx}^{c}|$ (where $x$ is the velocity direction) because of particle layering in the wall region.

A fictitious domain approach with Lagrange multiplier for fluid-structure interactions

We study a recently introduced formulation for fluid-structure interaction problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. The time discretization of the problem leads to a mixed problem for which a rigorous stability analysis is provided. The finite element space discretization is discussed and optimal convergence estimates are proved.

Numerical studies of the hysteresis in locomotion of a passively pitching foil

The direct-forcing fictitious domain method is extended to simulate the locomotion of a passively pitching foil. Our study focuses on the hysteresis phenomenon that the critical frequency for the reverse of the locomotion direction of the wing in case of decreasing frequency is smaller than that in case of increasing frequency. In our simulations, the hysteresis phenomenon is produced by imposing different initial conditions at a same frequency. Our results indicate that the ratio of the heaving amplitudes of two foil edges is crucial to the direction of the foil's horizontal motion, and the amplitude of the leading edge is generally smaller. The critical frequencies for the reverse of the locomotion direction are increased, when the foil-fluid density ratio is decreased or the spring constant is increased. The critical frequencies in the bi-stability regime also depend on the initial velocity imposed, and the hysteresis loop generally becomes larger if the initial velocities are closer to the terminal locomotion velocities of the foil.

Prediction of apparent properties with uncertain material parameters using high‐order fictitious domain methods and PGD model reduction

SummaryThis contribution presents a numerical strategy to evaluate the effective properties of image‐based microstructures in the case of random material properties. The method relies on three points: (1) a high‐order fictitious domain method; (2) an accurate spectral stochastic model; and (3) an efficient model‐reduction method based on the proper generalized decomposition in order to decrease the computational cost introduced by the stochastic model. A feedback procedure is proposed for an automatic estimation of the random effective properties with a given confidence. Numerical verifications highlight the convergence properties of the method for both deterministic and stochastic models. The method is finally applied to a real 3D bone microstructure where the empirical probability density function of the effective behaviour could be obtained. Copyright © 2016 John Wiley & Sons, Ltd.

Macroscopic characteristics of heavy particle resuspension obtained from direct numerical simulations of pressure driven channel flow

Three-dimensional direct numerical simulations are presented of the flow of fluid containing heavy spherical particles in a channel. The simulations resolve the motion of individual particles and the details of the flow at the particle level. They are based on the finite element method and the Distributed Lagrange Multiplier/ Fictitious Domain Method. The problem is characterized by four parameters, namely a Reynolds and an Archimedes number, a density ratio and a dimensionless particle size. The last three are fixed and the effects of the Reynolds number are studied over a range covering the onset of particle resuspension and the transition of flow to significant unsteadiness. Encouraging estimates for the onset of resuspension and the particle flux as a function of the excess Shields number are found. The results are analyzed in terms of the vertical distribution of horizontally averaged macroscopic quantities, obtained by applying integral momentum balances. On the basis of such balances, particle stresses and interfacial forces between the fluid and particle phases are estimated and compared to existing models, developed mainly for Stokes flow conditions. The qualitative agreement is encouraging but points to the need for taking into account several effects, notwithstanding those of finite particle Reynolds numbers. Provided that such information becomes available the set of four equations derived from the suspension and particle phase momentum equations could describe the average equilibrium distribution, the flow of both phases and the pressure in the fluid.

Unsteady flow of shear-thinning non-Newtonian incompressible fluid through a twin-screw extruder

The unsteady flow of viscous incompressible shear-thinning non-Newtonian fluid with mixed boundary is investigated. The boundary condition on the outflow is the modified natural boundary condition, it contains the additional nonlinear term, which enables us to control the kinetic energy of the backward flow. The global existence of weak solution is proved. The fictitious domain method which consists in filling the moving rigid screws with the surrounding fluid and taking into account the boundary conditions on these bodies by introducing a well-chosen distribution of boundary forces is used.

A 3D, fully Eulerian, VOF-based solver to study the interaction between two fluids and moving rigid bodies using the fictitious domain method

We present a three-dimensional (3D) and fully Eulerian approach to capturing the interaction between two fluids and moving rigid structures by using the fictitious domain and volume-of-fluid (VOF) methods. The solid bodies can have arbitrarily complex geometry and can pierce the fluid–fluid interface, forming contact lines. The three-phase interfaces are resolved and reconstructed by using a VOF-based methodology. Then, a consistent scheme is employed for transporting mass and momentum, allowing for simulations of three-phase flows of large density ratios. The Eulerian approach significantly simplifies numerical resolution of the kinematics of rigid bodies of complex geometry and with six degrees of freedom. The fluid–structure interaction (FSI) is computed using the fictitious domain method. The methodology was developed in a message passing interface (MPI) parallel framework accelerated with graphics processing units (GPUs). The computationally intensive solution of the pressure Poisson equation is ported to GPUs, while the remaining calculations are performed on CPUs. The performance and accuracy of the methodology are assessed using an array of test cases, focusing individually on the flow solver and the FSI in surface-piercing configurations. Finally, an application of the proposed methodology in simulations of the ocean wave energy converters is presented.

Numerical studies on the dynamics of an open triangle in a vertically oscillatory flow

A direct-forcing fictitious domain method is employed to study the dynamics of an open triangle in a vertically oscillatory flow. The flow structures, the vertical force and the torque on the fixed body are analysed for the stable flow regime in which the flow structures form and evolve exactly in the same way in each period and the unstable regime, respectively. Our results indicate that in the stable flow regime for the body with upright orientation, the steady streaming structure mainly comprises two vortex pairs located respectively above and below the body. Due to up–down asymmetry of the body, the pair below the body produces a larger vertical force on the body than the upper pair, which is mainly responsible for the non-zero average force at relatively high Reynolds numbers. The average force increases with increasing Reynolds number or increasing dimensionless period for the parameter range studied, due to the vortex effects. In the unstable regime, a vortex pair is ejected downward from each body edge. The irregular motion of the emitted vortices below the body leads to the irregular fluctuation of the vertical force. Regarding the torque on a tilted body, in the stable regime, the body experiences a restoring torque when its vertex angle is larger than a critical value being close to (and smaller than) 60°, and otherwise a destructive torque, irrespective of the value of tilt angle. For a fixed vertex angle, the torque magnitude is largest when the tilt angle is around 45°. In the unstable regime, the persistent ejection of the vortex pair during upward flow and corresponding restoring torque are observed for a large tilt angle with one edge aligned close to the horizontal direction, as in the experiment of Liu et al. (Phys. Rev. Lett., vol. 108, 2012, 068103). For a relatively small tilt angle, the emission direction of the vortex pair has intermittency, leading to the intermittency in the direction of torque. The reasons for the above observations are discussed. The predictions on the stable orientation for a hovering body in the stable flow regime and the irregular orientation in the unstable regime are confirmed in the dynamic simulation of a freely moving body. The body with the stable horizontal orientation in case of small vertex angle migrates along the body-shape-diverging direction.

A parallel fictitious domain method for the interface-resolved simulation of particle-laden flows and its application to the turbulent channel flow

ABSTRACTA parallel direct-forcing (DF) fictitious domain (FD) method for the simulation of particulate flows is reported in this paper. The parallel computing strategies for the solution of flow fields and particularly the distributed Lagrange multiplier are presented, and the high efficiency of the parallel code is demonstrated. The new code is then applied to study the effects of particle density (or particle inertia) on the turbulent channel flow. The results show that the large-scale vortices are weakened more severely, and the flow friction drag increases first and then reduces, as particle inertia is increased.

306 仮想領域有限要素法を用いた物体周りの流れ解析による流体力の評価(OS4-2 次世代シミュレーション技術の開拓)

When the fluid analysis is carried out in the case that there is a moving object in the flow field, the mesh regeneration method is applied due to the adaptive mesh refinement method. In addition, the shear-slip mesh update method is employed in case of rotating body problems. On the other hand, a method using the flow domain overlapped with the moving object is also proposed. In the finite difference method, a sub-grid is represented by moving object, and a main-grid is used for flow field calculation. The overset-grid is applied to the fluid analysis to reflect the sub-grid to main-grid by the least-squares method. Furthermore, the immersed boundary method based on the finite volume method is also employed. In the finite element analysis, the fictitious domain method is often adopted for the moving body problems. In this method, the computational domain is divided into foreground domain and background domain. In addition, the formulation is carried out based on the Lagrange multiplier method, and is applied to consider the velocity condition in the foreground domain. The finite element fluid analysis is carried out to reflect the physical quantity of the foreground domain to background domain. Physical quantities at any points of the foreground domain can be obtained from the physical quantities at each node by interpolation method. In this study, the flow field analysis using the fictitious domain finite element method is carried out.