Fictitious Domain Method Research Articles

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.

Rheology of sheared suspensions of rough frictional particles

AbstractThis paper presents three-dimensional numerical simulations of non-Brownian concentrated suspensions in a Couette flow at zero Reynolds number using a fictitious domain method. Contacts between particles are modelled using a discrete element method (DEM)-like approach, which allows for a more physical description, including roughness and friction. This work emphasizes the effect of friction between particles and its role on rheological properties, especially on normal stress differences. Friction is shown to notably increase viscosity and second normal stress difference $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}|N_2|$ and decrease $|N_1|$, in better agreement with experiments. The hydrodynamic and contact contributions to the overall particle stress are particularly investigated. This shows that the effect of friction is mostly due to the additional contact stress since the hydrodynamic stress remains unaffected by friction. Simulation results are also compared with experiments, such as normal stresses or effective friction coefficient $\mu (I_v)$, and the agreement is improved when friction is accounted for. This suggests that friction is operative in actual suspensions.

A study on unfitted 1D finite element methods

In the present paper we consider a 1D Poisson model characterized by the presence of an interface, where a transmission condition arises due to jumps of the coefficients. We aim at studying finite element methods with meshes not fitting such an interface. It is well known that when the mesh does not fit the material discontinuities the resulting scheme provides in general lower order accurate solutions. We focus on so-called embedded approaches, frequently adopted to treat fluid–structure interaction problems, with the aim of recovering higher order of approximation also in presence of non fitting meshes; we implement several methods inspired by: the Immersed Boundary method, the Fictitious Domain method, and the Extended Finite Element method. In particular, we present four formulations in a comprehensive and unified format, proposing several numerical tests and discussing their performance. Moreover, we point out issues that may be encountered in the generalization to higher dimensions and we comment on possible solutions.

Direct Numerical Simulation of Multiple Particles Sedimentation at an Intermediate Reynolds Number

AbstractIn this work the previously developed Lattice Boltzmann-Direct Forcing/ Fictitious Domain (LB-DF/FD) method is adopted to simulate the sedimentation of eight circular particles under gravity at an intermediate Reynolds number of about 248. The particle clustering and the resulting Drafting-Kissing-Tumbling (DKT) motion which takes place for the first time are explored. The effects of initial particle-particle gap on the DKT motion are found significant. In addition, the trajectories of particles are presented under different initial particle-particle gaps, which display totally three kinds of falling patterns provided that no DKT motion takes place, i.e. the concave-down shape, the shape of letter “M” and “in-line” shape. Furthermore, the lateral and vertical hydrodynamic forces on the particles are investigated. It has been found that the value of Strouhal number for all particles is the same which is about 0.157 when initial particle-particle gap is relatively large. The wall effects on falling patterns and particle expansions are examined in the final.

The finite and spectral cell methods for smart structure applications: transient analysis

This article introduces a robust and efficient numerical tool that is well suited for the simulation of ultrasonic guided waves and can be especially helpful when dealing with heterogeneous materials. The proposed method is based on a combination of high-order finite element methods (FEM) and the fictitious domain concept. If hierarchic shape functions, which are familiar from the p-version of the finite element method (p-FEM), are deployed the method is referred to as the finite cell method (FCM). Where Lagrange polynomials through Gaus–Lobatto–Legendre points are used, we refer to it as the spectral cell method (SCM). The name, SCM, derives from the fact that the deployed shape functions are commonly utilized in the spectral element method. To model smart structure applications such as shape control problems, noise cancelation devices and the excitation/sensing of ultrasonic guided waves, a coupling between electrical and mechanical variables is also taken into account. In this context, we focus on including the linear theory of piezoelectricity in the variational formulation of the FCM and the SCM, respectively. Several numerical benchmark problems are then used to validate the proposed approach. The simulations show promising results with respect to the accuracy of the method and the computational effort. We observe similar convergence properties for the proposed high-order fictitious domain methods as with “conventional” high-order finite element approaches. Implementing the proposed method in existing finite element software is, moreover, a straightforward process. These properties make the method an efficient tool for practical applications in structural health monitoring problems and smart structure applications in general.

The Isogeometric Shape Optimization Method Based on Finite Cell Method

Isogeometric analysis (IGA) method uses the same mathematical model in geometric design and engineering analysis, and is the most potential method to realize high accuracy and integrated optimization design. Finite Cell Method (FCM) introduces high order finite element method into fictitious domain method, and it has great advantages of complex boundary representation and high efficient convergence. In order to break through IGA’s limit on geomerty’s topology, IGA and FCM are combined together in this research. Function Heasivide and Dirac are used to approximate the computational domain and its first order derivative, then the stiffness matrix on the fictitious domain are calculated and the displacement in the shape with complex boundary is solved by IGA method. One order and two order implicit curves are used to the outer boundary representation of elastic optimization problem, and their coefficients are taken as design variable. The sensitivity formulas are deduced. MMA method is used to implement the IGA shape optimization based on FCM. The examples show that our method is efficient and the result is satisfied.

On a fictitious domain method with distributed Lagrange multiplier for interface problems

In this paper we propose a new variational formulation for an elliptic interface problem and discuss its finite element approximation. Our formulation fits within the framework of fictitious domain methods with distributed Lagrange multipliers. For the underlying mixed scheme we prove stability and convergence. Some preliminary numerical tests confirm the theoretical investigations.

Projected Krylov Methods for Solving Non-Symmetric Two-by-Two Block Linear Systems Arising from Fictitious Domain Formulations

The paper deals with the solution of large non-symmetric two-by-two block linear systems with a singular leading submatrix. Our algorithm consists of two levels. The outer level combines the Schur complement reduction with the orthogonal projectors that leads to the linear equation on subspaces. To solve this equation, we use a Krylov-type method representing the inner level of the algorithm. We propose a general technique how to get from the standard Krylov methods their projected variants generating iterations on subspaces. Then we derive the projected GMRES. The efficiency of our approach is illustrated by examples arising from the combination of the fictitious domain and FETI method.

Fully resolved simulation of particle deposition and heat transfer in a differentially heated cavity

In this paper a fictitious domain method is used to study the motion of particles in a differentially heated cavity. A collision strategy is implemented which is validated using the problem of two freely falling particles with natural convection taking place from the leading hot particle. The motion of the particles in a differentially heated cavity is considered where the vertical walls are subject to a temperature difference ΔT whereas horizontal walls are assumed to be adiabatic. Depending on the fluid Grashof number different flow regimes and two critical Grashof numbers are identified. Sustained motion of the suspended particles is also studied and different behaviour is observed compared to the limiting case of tracer particles where simulations are usually performed using one-way coupled point-particle assumptions. Finally the effects of the particles on the heat transfer from the hot wall are studied and it is found that addition of large particles can adversely influence the heat transfer rate. However, if hot particles are effectively removed from the wall, e.g. by increasing the Grashof number, wall heat transfer properties can still be enhanced.

A fictitious domain method for lithosphere‐asthenosphere interaction: Application to periodic slab folding in the upper mantle

AbstractWe present a new approach for the lithosphere‐asthenosphere interaction in subduction zones. The lithosphere is modeled as a Maxwell viscoelastic body sinking in the viscous asthenosphere. Both domains are discretized by the finite element method, and we use a staggered coupling method. The interaction is provided by a nonmatching interface method called the fictitious domain method. We describe a simplified formulation of this numerical technique and present 2‐D examples and benchmarks. We aim at studying the effect of mantle viscosity on the cyclicity of slab folding at the 660 km depth transition zone. Such cyclicity has previously been shown to occur depending on the kinematics of both the overriding and subducting plates, in analog and numerical models that approximate the 660 km depth transition zone as an impenetrable barrier. Here we applied far‐field plate velocities corresponding to those of the South‐American and Nazca plates at present. Our models show that the viscosity of the asthenosphere impacts on folding cyclicity and consequently on the slab's dip as well as the stress regime of the overriding plate. Values of the mantle viscosity between 3 and 5 × 1020 Pa s are found to produce cycles similar to those reported for the Andes, which are of the order of 30–40 Myr (based on magmatism and sedimentological records). Moreover, we discuss the episodic development of horizontal subduction induced by cyclic folding and, hence, propose a new explanation for episodes of flat subduction under the South‐American plate.

Coupling wavelets/vaguelets and smooth fictitious domain methods for elliptic problems: the univariate case

This work is devoted to the definition, the analysis and the implementation in the univariate case of a new numerical method for the approximation of partial differential equations solutions defined on complex domains. It couples a smooth fictitious domain method derived in Haslinger et al. (Numer Linear Algebra 14:713–739, 2007) with multiscale approximations. After the definition of the method, error estimates are derived: they allow to control a global error (on the whole domain including the boundary of the initial complex domain) as well as an interior error (for any sub-domain strictly included in the control domain). Numerical implementation and tests on univariate elliptic problems are finally described.

Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem

We extend our results on fictitious domain methods for Poisson’s problem to the case of incompressible elasticity, or Stokes’ problem. The mesh is not fitted to the domain boundary. Instead boundar ...

A fictitious domain method using equilibrated basis functions for harmonic and bi-harmonic problems in physics

In this paper we present a new meshless method to solve well-known problems in physics and engineering with either constant or variable material properties in 2D space. Harmonic and bi-harmonic problems are considered in this paper. The method constructs a set of basis functions, called Equilibrated Basis Functions (EqBFs), through a weighted residual integration over a fictitious domain embedding the main one. The bases can satisfy the governing partial differential equations approximately. Chebyshev polynomials of the first kind are employed to construct the EqBFs and exponential functions are used as the weights in the integrals. The parameters of the solution are arranged so that all the integrals can be decomposed into much simpler 1D ones over a normalized intervals. This reduces the computational efforts significantly. Either the EqBFs or the results of the integration process may be stored for further use. The validity of the results is examined through an extended patch test. A set of physical sample problems with variable/non-variable material properties; as the potential flow over a cylinder, steady-state heat conduction problems in an anisotropic inhomogeneous functionally graded material, potential and stokes flow through a channel of finite width obstructed by periodic cylinders, and the bending of thin elastic plates having constant or variable thickness are solved to demonstrate the capabilities of the method. As a preliminary study, we show that the method may effectively be used in a domain decomposition approach.

A fast algorithm for 3D simulation of thermal stratification in containment pools of nuclear power plants

Safety analysis is of ultimate importance for the operation of Nuclear Power Plants (NPP). Experiments even on a pilot scale are quite expensive and therefore numerical simulations play a major role in this analysis. They allow to predict the behaviour of NPP systems under different operational and accident conditions and to develop proper action plans for minimizing the risks of accidents, and minimizing the consequences of possible accidents. For a proper risk assessment and development of emergency plans, a very large number of scenarios have to be simulated, achieving acceptable accuracy for the critical parameters, such as radioactive pollution and temperature. The existing software tools are either very simplistic and therefore quite inaccurate or they use general CFD codes that makes them very slow. This paper presents a customized algorithm and corresponding software tools for simulation of non-isothermal flows in the containment pool of a NPP. It first summarizes the requirements for such simulation tools and then presents the new algorithm which, in the opinion of the authors, is a proper compromise between accuracy and efficiency for solving such problems. It uses a Fictitious Domain Method (FDM) to impose the boundary conditions, a Cartesian finite volume discretization combined with a Domain Decomposition (DD) technique, and a factorized perturbation of the incompressibility constraint proposed recently in Guermond and Minev (2010). Finally, results from numerical simulations in idealized and realistic geometries are presented and discussed.

A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem

We present a novel finite element method for the Stokes problem on fictitious domains. We prove inf-sup stability, optimal order convergence and uniform boundedness of the condition number of the discrete system. The finite element formulation is based on a stabilized Nitsche method with ghost penalties for the velocity and pressure to obtain stability in the presence of small cut elements. We demonstrate for the first time the applicability of the Nitsche fictitious domain method to three-dimensional Stokes problems. We further discuss a general, flexible and freely available implementation of the method and present numerical examples supporting the theoretical results.

Effects of the particle deformability on the critical separation diameter in the deterministic lateral displacement device

AbstractDeterministic lateral displacement (DLD) technology is a newly developed method which can separate microscale and nanoscale particles continuously and efficiently. In this paper, a direct numerical simulation method (i.e. a fictitious domain method) is used to simulate the motion of an elastic particle (modelled as homogeneously elastic body) in the DLD device. The effects of the particle deformability on the critical separation diameter are investigated. Our results indicate that there exists a critical deformability, below which the critical diameter decreases with increasing deformability, whereas beyond which the critical diameter increases with increasing deformability. The reasons are discussed via the consideration of the effects of the particle deformation and the lubrication force on the lateral position of the particle centre point. In addition, our results show that the increase in the gap distance between adjacent posts in both directions or in the longitudinal direction alone leads to the increase in the critical particle size with respect to the gap size, which can be explained by the lateral position of the separation streamline of the undisturbed flow.

Parallel Resolved Open Source CFD-DEM: Method, Validation and Application

In the following paper the authors present a fully parallelized Open Source method for calculating the interaction of immersed bodies and surrounding fluid. A combination of computational fluid dynamics (CFD) and a discrete element method (DEM) accounts for the physics of both the fluid and the particles. The objects considered are relatively big compared to the cells of the fluid mesh, i.e. they cover several cells each. Thus this fictitious domain method (FDM) is called resolved. The implementation is realized within the Open Source framework CFDEMcOupling ( www.cfdem.com ), which provides an interface between OpenFOAM® based CFD-solvers and the DEM software LIGGGHTS ( www.liggghts.com ). While both LIGGGHTS and OpenFOAM® were already parallelized, only a recent improvement of the algorithm permits the fully parallel computation of resolved problems. Alongside with a detailed description of the method, its implementation and recent improvements, a number of application and validation examples is presented in the scope of this paper.

Finite Element/Fictitious Domain programming for flows with particles made simple

Flows with suspended particles is a challenging task and important in many applications such as sedimentation, rheology and fluidized suspensions. The coupling between the suspending liquid flow and the particles’ motion is the central point in the complete understanding of the phenomena that occur in these applications. Finite Element/Fictitious Domain is an important class of method used to solve this problem. In this work we propose a simple object oriented implementation for simulations of flows with suspended particles in the plane using the Fictitious Domain method together with Lagrange multipliers to solve the Navier–Stokes and rigid body equations with a fully implicitly and fully coupled Finite Element approach. To have an efficient implementation for Fictitious Domain/Finite Element method, we introduce a new topological data structure that is concise in terms of storage and very suitable for searching the elements of the mesh intersected by the particles.

Computational simulation of the interactions between moving rigid bodies and incompressible two-fluid flows

We present a two-dimensional computational flow solver for simulation of two-way interactions between moving rigid bodies and two-fluid flows. The fluids are assumed to be incompressible and immiscible. The two-step projection method along with Graphics Processing Unit (GPU) acceleration is employed to solve the flow equations. The fluid–solid interaction is captured by using the fictitious domain method. A consistent mass and momentum scheme is implemented, which allows for simulation of multiphase flows characterized by large density ratios. The evolution of interfaces in the three-phase system is tracked by using the volume-of-fluid method with two scalar functions, representing the solid domain and one of the fluids. A geometrical approach is employed to reconstruct the interfaces in cells containing three phases and capture the intersection of phase interfaces (triple point). The performance and accuracy of the flow solver are assessed through a set of canonical test cases. Then, it is used to simulate the interactions between a free-floating buoy and waves generated by a bottom-hinged paddle in a wave tank.

A Preconditioner for Fictitious Domain Formulations of Elliptic PDEs on Uncertain Parameterized Domains

We consider the numerical solution of elliptic boundary-value problems on uncertain two-dimensional domains via the fictitious domain method. This leads to variational problems of saddle point form. Working under the standard assumption that the domain can be described by a finite number of independent random variables, discretization is achieved by a stochastic collocation mixed finite element method. We focus on the efficient iterative solution of the resulting sequence of indefinite linear systems and introduce a novel and efficient preconditioner for use with the minimal residual method. The challenging task is to construct a matrix that provides a robust approximation to a discrete representation of a trace space norm on a parameterized boundary.