Fictitious Domain Approach Research Articles

A local projection stabilization of fictitious domain method for elliptic boundary value problems

In this paper, a new consistent method based on local projections for the stabilization of a Dirichlet condition is presented in the framework of finite element method with a fictitious domain approach. The presentation is made on the Poisson problem but the theoretical and numerical results can be straightforwardly extended to any elliptic boundary value problem. A numerical comparison is performed with the Barbosa–Hughes stabilization technique. The advantage of the new stabilization technique is to affect only the equation on multipliers and thus to be equation independent.

A fictitious domain approach for the simulation of dense suspensions

Low Reynolds number concentrated suspensions do exhibit an intricate physics which can be partly unraveled by the use of numerical simulation. To this end, a Lagrange multiplier-free fictitious domain approach is described in this work. Unlike some methods recently proposed, the present approach is fully Eulerian and therefore does not need any transfer between the Eulerian background grid and some Lagrangian nodes attached to particles. Lubrication forces between particles play an important role in the suspension rheology and have been properly accounted for in the model. A robust and effective lubrication scheme is outlined which consists in transposing the classical approach used in Stokesian Dynamics to our present direct numerical simulation. This lubrication model has also been adapted to account for solid boundaries such as walls. Contact forces between particles are modeled using a classical Discrete Element Method (DEM), a widely used method in granular matter physics. Comprehensive validations are presented on various one-particle, two-particle or three-particle configurations in a linear shear flow as well as some O(103) and O(104) particle simulations.

A fictitious domain approach for the Stokes problem based on the extended finite element method

SUMMARYIn the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by extended finite element method and studied for the Poisson problem in a paper of Renard and Haslinger of 2009. The method allows computations in domains whose boundaries do not match. A mixed FEM is used for the fluid flow. The interface between the fluid and the structure is localized by a level‐set function. Dirichlet boundary conditions are taken into account using Lagrange multiplier. A stabilization term is introduced to improve the approximation of the normal trace of the Cauchy stress tensor at the interface and avoid the inf‐sup condition between the spaces for the velocity and the Lagrange multiplier. Convergence analysis is given, and several numerical tests are performed to illustrate the capabilities of the method. Copyright © 2013 John Wiley & Sons, Ltd.

A Fast Algorithm to Predict Cell Trajectories in Microdevices Using Dielectrophoresis

Prediction of accurate trajectories of biological particles is necessary for efficient design of microdevices for dielectrophoretic manipulation. Due to simplicity, a point-based method is generally used to simulate particle trajectory, but a point-based method provides significant distortion from the actual path when particle size is comparable to the device characteristic dimension. This article reports an efficient numerical model which can overcome these drawbacks of a point-based method and can be used for accurate predictions of particle trajectory. This model is formulated on a distributed Lagrange multiplier based fictitious domain (FD) approach for flow field and motion of particles, and the multi-domain method for electric potential. In this study, the dielectric forces are calculated from the Maxwell stress tensor. The accuracy of the proposed methods are validated separately with two test problems: terminal velocity of a particle in a stationary fluid while it is pulled with a constant force, and dielectrophoretic force acting on a particle when it is placed in between two planar electrodes. The capability of the proposed model is demonstrated by simulating trajectories of two biological particles (cells) of the same geometry and size but different dielectric properties in a microdevice. The effects of frequency, particle-particle initial separation distance, and particles’ relative position are investigated. Numerical results indicate that this model can capture the physics of particle manipulation better than the conventional point-based method. Moreover, this algorithm reduces the computational time significantly, which is a major bottleneck in 3-D simulation.

Relative performance of body fitted and fictitious domain simulations of flow through fixed packed beds of spheres

The relative performance of two numerical approaches involving body conforming and non-conforming grids for simulating porescale flow in complex configurations of fixed packed beds of spheres at moderate pore Reynolds numbers (12⩽Re⩽600) is examined. In the first approach, an unstructured solver is used with tetrahedral meshes which conform to the boundaries of the porespace. In the second approach, a fictitious domain formulation is used which employs non-body conforming Cartesian grids and enforces the no-slip conditions on the pore boundaries implicitly through a rigidity constraint force. Sphere to sphere contact points, where the fluid gap between solid boundaries becomes infinitesimal, are not resolved by either approach, but this is shown to have a negligible effect on the local flow field at the Reynolds numbers considered. Detailed grid convergence studies of both steady and unsteady flow through simple cubic packings indicate that for a fixed level of uncertainty, significantly lower grid densities may be used with the fictitious domain approach which also does not require complex grid generation techniques. This translates into large savings for simulation of flow through realistic packed beds, which is shown by both analytic estimates and actual CPU timings. The applicability of the fictitious domain approach is demonstrated by simulating unsteady flow through a randomly packed bed of 51 spheres at a pore Reynolds number of 600. The results are used to examine the dominance of helical vortices in the porescale flow field.

Modelling and simulation of 2D stokesian Squirmers

Direct numerical simulations of the individual and collective dynamics of neutral squirmers are presented. “Squirmer” refers to a class of swimmers driven by prescribed tangential deformations at their surface, and “cneutral” means that the swimmer does not apply a force dipole on the fluid. The squirmer model is used in this article to describe self-propelled liquid droplets. Each swimmer is a fluid sphere in Stokes flow without radial velocity and with a prescribed tangential velocity, which is constant in time in the swimmer frame. The interaction between two or more swimmers is taken into account through the relaxation of their translational and angular velocities. The algorithm presented for solving the fluid flow and the motion of the liquid particles is based on a variational formulation written on the whole domain (including the external fluid and the liquid particles) and on a fictitious domain approach. The constraint on the tangential velocity of swimmers can be enforced using two different methods: penalty approach of the strain rate tensor on the particles domain, or a saddle-point formulation involving a Lagrange multiplier associated to the constraint. This leads to a minimization problem over unconstrained functional spaces that can be implemented straightforwardly in a finiteelement multi-purpose solver. In order to ensure robustness, a projection algorithm is used to deal with contacts between particles. Two-dimensional numerical simulations implemented with FreeFem++ are presented.

On the spectral accuracy of a fictitious domain method for elliptic operators in multi-dimensions

This work is a continuation of the authors efforts to develop high-order numerical methods for solving elliptic problems with complex boundaries using a fictitious domain approach. In a previous paper, a new method was proposed, based on the use of smooth forcing functions with identical shapes, mutually disjoint supports inside the fictitious domain and whose amplitudes play the role of Lagrange multipliers in relation to a discrete set of boundary constraints. For one-dimensional elliptic problems, this method shows spectral accuracy but its implementation in two dimensions seems to be limited to a fourth-order algebraic convergence rate. In this paper, a spectrally accurate formulation is presented for multi-dimensional applications. Instead of being specified locally, the forcing function is defined as a convolution of a mollifier (smooth bump function) and a Lagrange multiplier function (the amplitude of the bump). The multiplier function is then approximated by Fourier series. Using a Fourier Galerkin approximation, the spectral accuracy is demonstrated on a two-dimensional Laplacian problem and on a Stokes flow around a periodic array of cylinders. In the latter, the numerical solution achieves the same high-order accuracy as a Stokes eigenfunction expansion and is much more accurate than the solution obtained with a classical third order finite element approximation using the same number of degrees of freedom.

Crystals stirred up: 1. Direct numerical simulations of crystal settling in nondilute magmatic suspensions

This is the first paper in a two‐part series examining the fluid dynamics of crystal settling and flotation in the lunar magma ocean. A key challenge in constraining solidification processes is determining the ability of individual crystals to decouple from vigorous thermal convection and settle out or float. The goal of this paper is to develop a computational methodology capable of capturing the complex solid‐fluid interactions that determine settling and flotation. In the second paper, we use this computational approach to explore the conditions under which plagioclase feldspar would be able to buoyantly float and form the earliest crust on the Moon. The direct numerical method described in this paper relies on a fictitious domain approach and captures solid‐body motion in 2D and 3D with little overhead beyond single fluid calculations. The two main innovations of our numerical implementation of a fictitious domain approach are an analytical quadrature scheme, which increases accuracy and reduces computational expense, and the derivation of a multibody collision scheme. Advantages of this approach over previous simulations of crystal‐bearing magmatic suspensions include the following: (1) we fully resolve the two‐way interaction between fluid and solid phases, implying that crystals are not only passively advected in an ambient flow field but are also actively driving flow, and (2) we resolve the flow around each individual crystal without assuming specific settling speeds or drag coefficients. We present several benchmark problems and convergence tests to validate our approach.

An efficient integration technique for the voxel‐based finite cell method

SUMMARYThe finite cell method is a fictitious domain approach based on hierarchical Ansatz spaces of higher order. The method avoids time‐consuming and often error‐prone mesh‐generation and favorably exploits Cartesian grids to embed structures of complex geometry in a simple‐shaped computational domain thus shifting parts of the computational effort from mesh generation to the computation within the embedding finite cells of regular shape. This paper presents an effective integration approach for voxel‐based models of linear elasticity that drastically reduces the computational effort on cell level. The applied strategy allows the pre‐computation of an essential part of the cell matrices and vectors of higher order, representing stiffness and load, respectively. Several benchmark problems show the potential of the proposed method in particular for heterogeneous material properties as common in biomedical applications based on computer tomography scans. The applied strategy ensures a fast computation for time‐critical simulations and even allows user‐interactive simulations for models of moderate size at a high level of accuracy. Copyright © 2012 John Wiley & Sons, Ltd.

Simulation of Enhanced Deposition Due to Magnetic Field Alignment of Ellipsoidal Particles in a Lung Bifurcation

Aerosolized chemotherapy has been recognized as a potential treatment for lung cancer. The challenge of providing sufficient therapeutic effects without reaching dose-limiting toxicity levels hinders the development of aerosolized chemotherapy. This could be mitigated by increasing drug-delivery efficiency with a noninvasive drug-targeting delivery method. The purpose of this study is to use direct numerical simulations to study the resulting local enhancement of deposition due to magnetic field alignment of high aspect ratio particles. High aspect ratio particles were approximated by a rigid ellipsoid with a minor diameter of 0.5 μm and fluid particle density ratio of 1,000. Particle trajectories were calculated by solving the coupled fluid particle equations using an in-house micro-macro grid finite element algorithm based on a previously developed fictitious domain approach. Particle trajectories were simulated in a morphologically realistic geometry modeling a symmetrical terminal bronchiole bifurcation. Flow conditions were steady inspiratory air flow due to typical breathing at 18 L/min. Deposition efficiency was estimated for two different cases: [1] particles aligned with the streamlines and [2] particles with fixed angular orientation simulating the magnetic field alignment of our previous in vitro study. The local enhancement factor defined as the ratio between deposition efficiency of Case [1] and Case [2] was found to be 1.43 and 3.46 for particles with an aspect ratio of 6 and 20, respectively. Results indicate that externally forcing local alignment of high aspect ratio particles can increase local deposition considerably.

Coiflet–Galerkin method for solving second order BVPs with variable coefficients in three dimensions

Three dimensional BVPs with variable coefficients, need to evaluate a lot of complex integrals in Galerkin method solutions. We consider Coiflets, as basis functions to evaluate the complex integrals fast and effective, via three variate connection coefficients on the interval [0, 2n]. We show that these connection coefficients can be computed independent of n, by solving small linear systems. Moreover we prove that the rate of convergence of approximate solution to the exact solution, is O(2 − nN) where N is the degree of Coiflets. Besides, in our proposal we do not adopt fictitious domain approach which is commonly used in wavelet-Galerkin methods. In this way computational cost is reduced. To confirm the theoretical results, a numerical experiment is presented.

Small and large deformation analysis with the p- and B-spline versions of the Finite Cell Method

The Finite Cell Method (FCM) is an embedded domain method, which combines the fictitious domain approach with high-order finite elements, adaptive integration, and weak imposition of unfitted Diric...

The Finite Cell Method for large deformation analysis

AbstractThe Finite Cell Method (FCM) combines the fictitious domain approach with high‐order finite elements and adaptive integration. For linear elastic problems with smooth solution, FCM has been shown to achieve exponential rates of convergence in energy norm, while its structured cell grid guarantees simple mesh generation irrespective of the geometric complexity involved. In this contribution, the FCM idea is combined with standard finite element technology for the solution of geometrically nonlinear problems. In particular, a modified FCM formulation is introduced, which resets the deformed configuration of the fictitious domain to the deformation‐free reference configuration after each Newton iteration. Numerical experiments show that this intervention allows for stable nonlinear FCM analysis with very small values of the penalty parameter, while the accuracy of the geometrically nonlinear solution within the physical domain remains unaffected. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The hp‐d‐adaptive finite cell method for geometrically nonlinear problems of solid mechanics

SUMMARYThe finite cell method (FCM) combines the fictitious domain approach with the p‐version of the finite element method and adaptive integration. For problems of linear elasticity, it offers high convergence rates and simple mesh generation, irrespective of the geometric complexity involved. This article presents the integration of the FCM into the framework of nonlinear finite element technology. However, the penalty parameter of the fictitious domain is restricted to a few orders of magnitude in order to maintain local uniqueness of the deformation map. As a consequence of the weak penalization, nonlinear strain measures provoke excessive stress oscillations in the cells cut by geometric boundaries, leading to a low algebraic rate of convergence. Therefore, the FCM approach is complemented by a local overlay of linear hierarchical basis functions in the sense of the hp‐d method, which synergetically uses the h‐adaptivity of the integration scheme. Numerical experiments show that the hp‐d overlay effectively reduces oscillations and permits stronger penalization of the fictitious domain by stabilizing the deformation map. The hp‐d‐adaptive FCM is thus able to restore high convergence rates for the geometrically nonlinear case, while preserving the easy meshing property of the original FCM. Accuracy and performance of the present scheme are demonstrated by several benchmark problems in one, two, and three dimensions and the nonlinear simulation of a complex foam sample. Copyright © 2011 John Wiley & Sons, Ltd.

Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains

A tensor-based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains. It provides an accurate solution which is explicit with respect to parameters defining the shape of the domain, thus allowing efficient a posteriori probabilistic or parametric analyses. In the proposed method, a fictitious domain approach is first adopted for the reformulation of the parametric problem on a fixed domain, yielding a weak formulation in a tensor product space (product of space functions and parametric functions). The paper is limited to the case of Neumann conditions on uncertain parts of the boundary. The Proper Generalized Decomposition method is then introduced for the construction of a tensor product approximation (separated representation) of the solution. It can be seen as an a priori model reduction technique which automatically captures reduced bases of space functions and parametric functions which are optimal for the representation of the solution. This tensor-based method is made computationally tractable by introducing separated representations of variational forms, resulting from separated representations of the parameterized indicator function of the uncertain domain. For this purpose, a method is proposed for the construction of a constrained tensor product approximation which preserves positivity and therefore ensures well-posedness of problems associated with approximate indicator functions. Moreover, a regularization of the geometry is introduced to speed up the convergence of these tensor product approximations.

The finite cell method for bone simulations: verification and validation

Standard methods for predicting bone's mechanical response from quantitative computer tomography (qCT) scans are mainly based on classical h-version finite element methods (FEMs). Due to the low-order polynomial approximation, the need for segmentation and the simplified approach to assign a constant material property to each element in h-FE models, these often compromise the accuracy and efficiency of h-FE solutions. Herein, a non-standard method, the finite cell method (FCM), is proposed for predicting the mechanical response of the human femur. The FCM is free of the above limitations associated with h-FEMs and is orders of magnitude more efficient, allowing its use in the setting of computational steering. This non-standard method applies a fictitious domain approach to simplify the modeling of a complex bone geometry obtained directly from a qCT scan and takes into consideration easily the heterogeneous material distribution of the various bone regions of the femur. The fundamental principles and properties of the FCM are briefly described in relation to bone analysis, providing a theoretical basis for the comparison with the p-FEM as a reference analysis and simulation method of high quality. Both p-FEM and FCM results are validated by comparison with an in vitro experiment on a fresh-frozen femur.

Shell Finite Cell Method: A high order fictitious domain approach for thin-walled structures

This article presents a generalization of the recently proposed Finite Cell Method to thin-walled structures. This approach uses a combination of well known Fictitious Domain Methods with high order hierarchical Ansatz spaces known from the p-version of the Finite Element Method. Whereas the original concept embeds a three-dimensional structure in a simple domain being meshed into a grid of cube shaped cells, the extension presented in this paper applies the fictitious domain approach to a two-dimensional master domain defined in the parameter plane of the geometric model. Implementation details are discussed and numerical benchmark problems show the high accuracy and computational efficiency of the new approach. It is also remarked, that the present approach can easily be carried over to isogeometric analysis, opening an attractive possibility to simulate trimmed NURBS-surfaces.

Fictitious domain methods for two-phase flow energy balance computations in nuclear components

This paper is dedicated to the numerical simulation of nuclear components (cores and steam generators) by fictitious domain methods. The fictitious domain approach consists in immersing the physical domain under study in a Cartesian domain, called the fictitious domain, and in performing the numerical resolution on this fictitious domain. The calculation times are then efficiently reduced by the use of fast solvers. In counterpart, one has to handle with an immersed boundary, generally non-aligned with the Cartesian mesh, which can be non-trivial. The two fictitious domain methods compared here on industrial simulations and developed by Ramière et al. deal with an approximate immersed interface directly derived from the uniform Cartesian mesh. All the usual immersed boundary conditions (Dirichlet, Robin, Neumann), possibly mixed, are handled through a unique formulation of the fictitious problem. This kind of approximation leads to first-order methods in space that exhibit a good ratio of the precision of the approximate solution over the CPU time, which is very important for industrial simulations. After a brief recall of the fictitious domain method with spread interface (Ramière et al., CMAME 2007) and the fictitious domain method with immersed jumps (Ramière et al., JCP 2008), we will focus on the numerical results provided by these methods applied to the energy balance equation in a steam generator. The advantages and drawbacks of each method will be pointed out. Generally speaking, the two methods confirm their very good efficiency in terms of precision, convergence, and calculation time in an industrial context. Copyright © 2011 John Wiley & Sons, Ltd.

Non-standard bone simulation: interactive numerical analysis by computational steering

Numerous numerical methods have been developed in an effort to accurately predict stresses in bones. The largest group are variants of the h-version of the finite element method (h-FEM), where low order Ansatz functions are used. By contrast, we3 investigate a combination of high order FEM and a fictitious domain approach, the finite cell method (FCM). While the FCM has been verified and validated in previous publications, this article proposes methods on how the FCM can be made computationally efficient to the extent that it can be used for patient specific, interactive bone simulations. This approach is called computational steering and allows to change input parameters like the position of an implant, material or loads and leads to an almost instantaneous change in the output (stress lines, deformations). This direct feedback gives the user an immediate impression of the impact of his actions to an extent which, otherwise, is hard to obtain by the use of classical non interactive computations. Specifically, we investigate an application to pre-surgical planning of a total hip replacement where it is desirable to select an optimal implant for a specific patient. Herein, optimal is meant in the sense that the expected post-operative stress distribution in the bone closely resembles that before the operation.

Direct simulations of particle deposition and filtration in dual-scale porous media

A two dimensional direct numerical simulation technique is developed to describe particulate flows in dual-scale porous media to predict particle deposition on the permeable porous surface in liquid composite molding processes. This individual particle level simulation accounts for hydrodynamic interaction between particles and the fluid, especially near a porous wall (fiber tow), and can predict the deposition of the particles on solid or porous surfaces. A Stokes–Brinkman coupling is employed to describe the flow in dual-scale porous media and a fictitious domain approach is used to deal with freely suspended particles in the fluid stream. A single particle deposition process is investigated extensively along with effects of the permeability of porous media, the particle size and the pressure drop. Mechanisms leading to accelerated or delayed deposition of particles are analyzed by investigating the velocity fields around the particle in close proximity of the porous surface. Finally, particle filtration simulation are performed with a large number of particles to demonstrate the feasibility of this scheme to address particle deposition and filtration during manufacturing of composites using liquid composite molding processes in which the particles are mixed with the resin and the suspension is injected into a stationary dual scale preform.