NEW FICTITIOUS DOMAIN METHODS: FORMULATION AND ANALYSIS

Flow simulations in rotary volumetric pumps and compressors with the fictitious domain method

On the application of immersed boundary, fictitious domain and body-conformal mesh methods to many particle multiphase flows

The purpose of this paper is to analyze the fluid and moving boundary problem by fictitious domain method with Lagrange multiplier. The incompressible Navier-Stokes equation is used in this paper. To solve this equation, fictitious domain method with Lagrange multiplier is applied. For the analysis of the finite element method, the mixed interpolation based on the bubble function is presented. In fictitious domain method, this method is not necessary to remeshing. Additionally, the calculation time is decreased, and the amount of memory is reduced. It is necessary to approximate by using interpolation function for the Lagrange multiplier. However, it is dicult to integrate the formula. The Delta function of Dirac is used here. Two kinds of meshes are used for fictitious domain method. One of meshes is used as computational domain, the other is used as a bounded domain. In this research, it is analyzed to move the object by the fluid force (drag force) in 3-dimension. In fictitious domain method, it is easy to obtain the fluid force. Because it is defined the Lagrange multiplier is equal to the traction in each node on a bounded domain. The pressure and flow velocity when the object moves are verified. And advantages of fictitious domain method based on distributed Lagrange multiplier are investigated.

An improved bound is derived on the rate of convergence of the standard difference scheme for solving the Dirichlet problem for the Helmholtz equation in domains of arbitrary shape. The scheme is based on the fictitious domain method. An application of the fictitious domain method and the grid method for solving one nonlinear boundary-value problem is considered. A rate of convergence bound is obtained for the proposed method.

Fictitious domain method (FDM) is a commonly accepted direct numerical simulation technique for moving boundary problems. Indicator function used to distinguish the solid zone and the fluid zone is an essential part concerning the whole prediction accuracy of FDM. In this work, a new indicator function through volume intersection is developed for FDM. In this method, the arbitrarily polyhedral cells across the interface between fluid and solid are located and subdivided into tetrahedrons. The fraction of the solid volume in each cell is accurately computed to achieve high precision of integration calculation in the particle domain, improving the accuracy of the whole method. The quadrature over the solid domain shows that the newly developed indicator function can provide results with high accuracy for variable integration in both stationary and moving boundary problems. Several numerical tests, including flow around a circular cylinder, a single sphere in a creeping shear flow, settlement of a circular particle in a closed container, and in-line oscillation of a circular cylinder, have been performed. The results show good accuracy and feasibility in dealing with the stationary boundary problem as well as the moving boundary problem. This method is accurate and conservative, which can be a feasible tool for studying problems with moving boundaries.

Fictitious Domain approach with hp-finite element approximation for incompressible fluid flow

Fictitious domain methods facilitate the discretization of boundary value problems by applying simple meshes containing finite elements or cells that do not conform to the geometry of the domain of interest. In this way, the effort of meshing complex domains is shifted to the numerical integration of those elements/cells that are cut by the boundary of the domain. In this chapter, we will first introduce a high-order fictitious domain method and then present adaptive methods that are suited for the numerical integration of broken elements and cells. Since the quadrature schemes presented in this chapter are quite general, they can be applied to the different versions of fictitious domain methods.

Article Solution of the Stokes nonstationary problems by the fictitious domain method was published on January 1, 1995 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 10, issue 3).

Standard finite element methods employ an element‐wise assembly strategy. The element's contribution to the system matrix is formed by a loop over quadrature points. This concept is also used in fictitious domain methods, which perform simulations on a simple tensor‐product background mesh cut by a boundary representation that defines the domain of interest.Considering such d‐dimensional background meshes based on splines of degree p with maximal smoothness, Cp−1, the cost of setting up the system matrix is 𝒪(p3d) per degree of freedom. Alternative assembly and formation techniques can significantly reduce this cost. In particular, the combination of (1) sum factorization, (2) weighted quadrature, and (3) row‐based assembly yields a cost of 𝒪(pd+1) for non‐cut background meshes. However, applying this fast approach to cut background meshes is an open challenge since they do not have a tensor‐product structure.This work presents techniques that allow the treatment of cut background meshes and thus the application of fast formation and assembly to fictitious domain methods. First, a discontinuous version of weighted quadrature is presented, which introduces a discontinuity into a cut test function's support. The cut region can be treated separately from the non‐cut counterpart; the latter can be assembled by the fast concepts. A three‐dimensional example investigates the accuracy and efficiency of the proposed concept and demonstrates its speed‐up compared to conventional formation and assembly.

A new Galerkin finite element method for the solution of the Navier–Stokes equations in enclosures containing internal parts which may be moving is presented. Dubbed the virtual finite element method, it is based upon optimization techniques and belongs to the class of fictitious domain methods. Only one volumetric mesh representing the enclosure without its internal parts needs to be generated. These are rather discretized using control points on which kinematic constraints are enforced and introduced into the mathematical formulation by means of Lagrange multipliers. Consequently, the meshing of the computational domain is much easier than with classical finite element approaches. First, the methodology will be presented in detail. It will then be validated in the case of the two-dimensional Couette cylinder problem for which an analytical solution is available. Finally, the three-dimensional fluid flow inside a mechanically agitated vessel will be investigated. The accuracy of the numerical results will be assessed through a comparison with experimental data and results obtained with a standard finite element method. © 1997 John Wiley & Sons, Ltd.

In this article we discuss the fictitious domain solution of the Navier‐Stokes equations modelling unsteady incompressible viscous flow. The method is based on a Lagrange multiplier treatment of the boundary conditions to be satisfied and is particularly well suited to the treatment of no‐slip boundary conditions. This approach allows the use of structured meshes and fast specialized solvers for problems on complicated geometries. Another interesting feature of the fictitious domain approach is that it allows the solution of optimal shape problems without regriding. The resulting methodology is applied to the solution of flow problems including external incompressible viscous flow modelled by the Navier‐Stokes equations and then to an optimal shape problem for Stokes and Navier‐Stokes flow.

A smooth integration of geometric models and numerical simulation has been in the focus of research in computational mechanics for long, as the classical transition from CAD-based geometric models to finite element meshes is, despite all support by sophisticated preprocessors, very often still error prone and time consuming. High order finite element methods bear some advantages for a closer coupling, as much more complex surface types can be represented by p-elements than by the classical low order approach. Significant progress in the direction of model integration has recently been made with the introduction of the ‘iso-geometric analysis’ concept, where the discretization of surfaces and the Ansatz for the shape functions is based on a common concept of a NURBS-description. In this paper we discuss a recently proposed different approach, the Finite Cell Method, which combines ideas from fictitious domain methods with high order approximation techniques. The basic idea is an extension of a partial differential equation beyond the physical domain of computation up to the boundaries of an embedding domain, which can easier be meshed. The actual domain is only taken into account using a precise integration technique of ‘cells’ which are cut by the domains' boundary. If this extension is smooth, the solution can be well approximated by high order polynomials. The method shows exponential rate of convergence for smooth problems and good accuracy even in the presence of singularities.

Rotational test spaces for a fully-implicit FVM and FEM for the DNS of fluid-particle interaction

The hybrid spectral difference methods (HSD) for the Laplace and Helmholtz equations in exterior domains are proposed. We consider the fictitious domain method with the absorbing boundary conditions (ABCs). The HSD method is a finite difference version of the hybridized Galerkin method, and it consists of two types of finite difference approximations; the cell finite difference and the interface finite difference. The fictitious domain is composed of two subregions; the Cartesian grid region and the boundary layer region in which the radial grid is imposed. The boundary layer region with the radial grid makes it easy to implement the discrete radial ABC. The discrete radial ABC is a discrete version of the Bayliss–Gunzburger–Turkel ABC without pertaining any radial derivatives. Numerical experiments confirming efficiency of our numerical scheme are provided.

In the present work, we propose a new approach, the so-called compressed adaptive integration scheme (C-AIS), for the computation of the stiffness and mass matrices in fictitious domain methods requiring the integration of discontinuous functions. The novel approach extends the conventional quadtree-decomposition-based adaptive integration scheme (AIS) by an additional step, in which established image-compression techniques are exploited to decrease the number of integration sub-cells. The benefits of the C-AIS are manifold: First, the compression of the sub-cells inevitably leads to significant savings in terms of computational time required by the numerical integration. Second, the compression procedure, which is executed directly after the quadtree-decomposition algorithm, can be easily included in existing codes. Third, if applied to polynomial integrands, the C-AIS yields exactly the same accuracy as the conventional AIS. Finally, the fourth advantage is seen in the fact that the C-AIS can readily be combined with other approaches seeking a reduction of the number of integration points such as the Boolean-FCM. The efficiency of the C-AIS approach is presented in the context of the FCM based on Cartesian meshes applied to problems of linear elastostatics and modal analysis, while it is also suitable for the quadrature in other fictitious domain approaches, e.g., CutFEM and cgFEM.