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An extended pressure finite element space for twophase incompressible flows with surface tension
, 2007
"... Abstract. We consider a standard model for incompressible twophase flows in which a localized force at the interface describes the effect of surface tension. If a level set (or VOF) method is applied then the interface, which is implicitly given by the zero level of the level set function, is in ge ..."
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Abstract. We consider a standard model for incompressible twophase flows in which a localized force at the interface describes the effect of surface tension. If a level set (or VOF) method is applied then the interface, which is implicitly given by the zero level of the level set function, is in general not aligned with the triangulation that is used in the discretization of the flow problem. This nonalignment causes severe difficulties w.r.t. the discretization of the localized surface tension force and the discretization of the flow variables. In cases with large surface tension forces the pressure has a large jump across the interface. In standard finite element spaces, due to the nonalignment, the functions are continuous across the interface and thus not appropriate for the approximation of the discontinuous pressure. In many simulations these effects cause large oscillations of the velocity close to the interface, socalled spurious velocities. In [1] it is shown that an extended finite element space (XFEM) is much better suited for the discretization of the pressure variable. In this paper we derive important properties of the XFEM space. We present (optimal) approximation error bounds and prove that the diagonally scaled mass matrix has a uniformly bounded spectral condition number. Results of numerical experiments are presented that illustrate properties of the XFEM space.
Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations
, 2006
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AN EXTENDED FINITE ELEMENT METHOD APPLIED TO LEVITATED DROPLET PROBLEMS
"... Abstract. We consider a standard model for incompressible twophase flows in which a localized force at the interface describes the effect of surface tension. If a level set method is applied then the approximation of the interface is in general not aligned with the triangulation. This causes severe ..."
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Cited by 8 (0 self)
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Abstract. We consider a standard model for incompressible twophase flows in which a localized force at the interface describes the effect of surface tension. If a level set method is applied then the approximation of the interface is in general not aligned with the triangulation. This causes severe difficulties w.r.t. the discretization and often results in large spurious velocities. In this paper we reconsider a (modified) extended finite element method (XFEM), which in previous papers has been investigated for relatively simple twophase flow model problems, and apply it to a physically realistic levitated droplet problem. The results show that due to the extension of the standard FE space one obtains much better results in particular for large interface tension coefficients. Furthermore, a certain cutoff technique results in better efficiency without sacrificing accuracy.
FINITE ELEMENT METHODS ON VERY LARGE, DYNAMIC TUBULAR GRID ENCODED IMPLICIT SURFACES
"... Abstract. The simulation of physical processes on interfaces and a variety of applications in geometry processing and geometric modeling are based on the solution of partial differential equations on curved and evolving surfaces. Frequently, an implicit level set type representation of these surface ..."
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Cited by 6 (2 self)
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Abstract. The simulation of physical processes on interfaces and a variety of applications in geometry processing and geometric modeling are based on the solution of partial differential equations on curved and evolving surfaces. Frequently, an implicit level set type representation of these surfaces is the most effective and computationally advantageous approach. This paper addresses the computational problem of how to solve partial differential equations on highly resolved level sets with an underlying very highresolution discrete grid. These highresolution grids are represented in a very efficient Dynamic Tubular Grid encoding format for a narrow band. A reaction diffusion model on a fixed surface and surface evolution driven by a nonlinear geometric diffusion approach, by isotropic, or truly anisotropic curvature motion are investigated as characteristic model problems. The proposed methods are based on semiimplicit finite element discretizations directly on these narrow bands, require only standard numerical quadrature and allow for large time steps. To combine large time steps with a very thin and thus storage inexpensive narrow band, suitable transparent boundary conditions on the boundary of the narrow band and a nested iteration scheme in each time step are investigated. This nested iteration scheme enables the discrete interfaces to move in a single time step significantly beyond the domain of the narrow band of the previous time step. Furthermore, algorithmic tools are provided to assemble finite element matrices and to apply matrix vector operators via fast, cachecoherent access to the Dynamic Tubular Grid encoded data structure. The consistency of the presented approach is evaluated and various numerical examples show its application potential. Key words. level set methods, narrow band approach, partial differential equations on surfaces, curvature motion AMS subject classifications. 65D18, 65D10, 65M50 65M60 65N30, 65N50, 68P05, 68P20
Simple finite elementbased computation of distance functions in unstructured grids
"... A distance field is a representation of the closest distance from a point to a given surface. Distance fields are widely used in applications ranging from computer vision, physics and computer graphics and have been the subject of research of many authors in the last decade. Most of the methods for ..."
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A distance field is a representation of the closest distance from a point to a given surface. Distance fields are widely used in applications ranging from computer vision, physics and computer graphics and have been the subject of research of many authors in the last decade. Most of the methods for computing distance fields are devoted to Cartesian grids while little attention has been paid to unstructured grids. Finite element methods are well known for their ability to deal with partial differential equations in unstructured grids. Therefore, we propose an extension of the fast marching method for computing a distance field in a finite element context employing the element interpolation to hold the Eikonal property (‖∇φ‖=1). A simple algorithm to develop the computations is also presented and its efficiency demonstrated through various unstructured grid examples. We observed that the presented algorithm has processing times proportional
NITSCHE’S METHOD FOR A TRANSPORT PROBLEM IN TWOPHASE INCOMPRESSIBLE FLOWS
"... Abstract. We consider a parabolic interface problem which models the transport of a dissolved species in twophase incompressible flow problems. Due to the socalled Henry interface condition the solution is discontinuous across the interface. We use an extended finite element space combined with a ..."
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Abstract. We consider a parabolic interface problem which models the transport of a dissolved species in twophase incompressible flow problems. Due to the socalled Henry interface condition the solution is discontinuous across the interface. We use an extended finite element space combined with a method due to Nitsche for the spatial discretization of this problem and derive optimal discretization error bounds for this method. For the time discretization a standard θscheme is applied. Results of numerical experiments are given that illustrate the convergence properties of this discretization. Key words. Nitsche’s method, interface problem, extended finite elements, twophase flows, AMS subject classifications. 65N30, 65N40 1. Introduction. Let Ω ⊂ R d, d = 2, 3, be a convex polygonal domain that contains two different immiscible incompressible phases. The (in general time dependent) subdomains containing the two phases are denoted by Ω1, Ω2, with ¯ Ω = ¯ Ω1 ∪ ¯ Ω2. A typical example is a droplet surrounded by another fluid. In this paper we only consider the stationary case in which the interface Γ: = ¯ Ω1 ∩ ¯ Ω2 does not depend
FINITE ELEMENT DISCRETIZATION ERROR ANALYSIS OF A SURFACE TENSION FORCE IN TWOPHASE INCOMPRESSIBLE FLOWS∗
"... Abstract. We consider a standard model for a stationary twophase incompressible flow with surface tension. In the variational formulation of the model a linear functional which describes the surface tension force occurs. This functional depends on the location and the curvature of the interface. In ..."
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Abstract. We consider a standard model for a stationary twophase incompressible flow with surface tension. In the variational formulation of the model a linear functional which describes the surface tension force occurs. This functional depends on the location and the curvature of the interface. In a finite element discretization method the functional has to be approximated. For an approximation method based on a LaplaceBeltrami representation of the curvature we derive sharp bounds for the approximation error. A new modified approximation method with a significantly smaller error is introduced.
AN EULERIAN FINITE ELEMENT METHOD FOR ELLIPTIC EQUATIONS ON MOVING SURFACES
"... Abstract. In this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated. The main idea is to use finite element spaces that are induced by triangulations of an “outer ” domain to discretize the partial differential equation on th ..."
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Abstract. In this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated. The main idea is to use finite element spaces that are induced by triangulations of an “outer ” domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface, for example, twophase incompressible flow problems. We give an analysis that shows that the method has optimal order of convergence both in the H 1 and in the L 2norm. Results of numerical experiments are included that confirm this optimality. Key words. Surface, interface, finite element, level set method, twophase flow, Marangoni
The Nitsche XFEMDG spacetime method and its implementation in three space dimensions
 SIAM J. Sci. Comput
"... new finite element discretization method for a class of twophase mass transport problems is presented and analyzed. The transport problem describes mass transport in a domain with an evolving interface. Across the evolving interface a jump condition has to be satisfies. The discretization in that p ..."
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new finite element discretization method for a class of twophase mass transport problems is presented and analyzed. The transport problem describes mass transport in a domain with an evolving interface. Across the evolving interface a jump condition has to be satisfies. The discretization in that paper is a spacetime approach which combines a discontinuous Galerkin (DG) technique (in time) with an extended finite element method (XFEM). Using the Nitsche method the jump condition is enforced in a weak sense. While the emphasis in that paper was on the analysis and one dimensional numerical experiments the main contribution of this paper is the discussion of implementation aspects for the spatially three dimensional case. As the spacetime interface is typically given only implicitly as the zerolevel of a levelset function, we construct a piecewise planar approximation of the spacetime interface. This discrete interface is used to divide the spacetime domain into its subdomains. An important component within this decomposition is a new method for dividing fourdimensional prisms intersected by a piecewise planar spacetime interface into simplices. Such a subdivision algorithm is necessary for numerical integration on the subdomains as well as on the spacetime interface. These numerical integrations are needed in the implementation of the Nitsche XFEMDG method in three space dimensions. Corresponding numerical studies are presented and discussed.