A finite element based level set method for two-phase incompressible flows

We present a method that has been developed for the efficient numerical simulation of two-phase incompressible flows. For capturing the interface between the phases the level set technique is applied. The continuous model consists of the incompressible Navier–Stokes equations coupled with an advection equation for the level set function. The effect of surface tension is modeled by a localized force term at the interface (so-called continuum surface force approach). For spatial discretization of velocity, pressure and the level set function conforming finite elements on a hierarchy of nested tetrahedral grids are used. In the finite element setting we can apply a special technique to the localized force term, which is based on a partial integration rule for the Laplace–Beltrami operator. Due to this approach the second order derivatives coming from the curvature can be eliminated. For the time discretization we apply a variant of the fractional step θ-scheme. The discrete saddle point problems that occur in each time step are solved using an inexact Uzawa method combined with multigrid techniques. For reparametrization of the level set function a new variant of the fast marching method is introduced. A special feature of the solver is that it combines the level set method with finite element discretization, Laplace–Beltrami partial integration, multilevel local refinement and multigrid solution techniques. All these components of the solver are described. Results of numerical experiments are presented.     

Computation of incompressible bubble dynamics with a stabilized finite element level set method

This paper presents a stabilized finite element method for the three dimensional computation of incompressible bubble dynamics using a level set method. The interface between the two phases is resolved using the level set approach developed by Sethian [Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999], Sussman et al. [J. Comput. Phys. 114 (1994) 146], and Sussman et al. [J. Comput. Phys. 148 (1999) 81–124]. In this approach the interface is represented as a zero level set of a smooth function. The streamline-upwind/Petrov–Galerkin method was used to discretize the governing flow and level set equations. The continuum surface force (CSF) model proposed by Brackbill et al. [J. Comput. Phys. 100 (1992) 335–354] was applied in order to account for surface tension effects. To restrict the interface from moving while re-distancing, an improved re-distancing scheme proposed in the finite difference context [J. Comput. Phys. 148 (1999) 81–124] is adapted for finite element discretization. This enables us to accurately compute the flows with large density and viscosity differences, as well as surface tension. The capability of the resultant algorithm is demonstrated with two and three dimensional numerical examples of a single bubble rising through a quiescent liquid, and two bubble coalescence.     

An extended residual-based variational multiscale method for two-phase flow including surface tension

In this study, an extended residual-based variational multiscale method is proposed for two-phase flow including surface tension. The extended residual-based variational multiscale method combines a residual-based form of the variational multiscale method and the extended finite element method (XFEM). By extending the solution spaces, it is possible to reproduce discontinuities of the solution fields inside elements intersected by the interface. In particular, we propose a quasi-static enrichment to reproduce time-dependent discontinuities. Kink enrichments of both velocity and pressure as well as kink enrichments of velocity combined with jump enrichments of pressure are considered here. To capture the interface between the phases on a fixed grid, a level-set approach is used. A residual-based variational multiscale method is employed for computing both flow and interface motion. The presented method is tested for various two-phase flow examples exhibiting small and large density and viscosity ratios, with and without surface tension: a two-phase Couette flow, a Rayleigh–Taylor instability, a sloshing tank and a three-dimensional rising bubble. To the best of our knowledge, these are the first simulation results for representative time-dependent three-dimensional two-phase flow problems using an extended finite element method. Stable and accurate results are obtained for all test examples.     

Stabilized extended finite elements for the approximation of saddle point problems with unfitted interfaces

We address a two-phase Stokes problem, namely the coupling of two fluids with different kinematic viscosities. The domain is crossed by an interface corresponding to the surface separating the two fluids. We observe that the interface conditions allow the pressure and the velocity gradients to be discontinuous across the interface. The eXtended Finite Element Method (XFEM) is applied to accommodate the weak discontinuity of the velocity field across the interface and the jump in pressure on computational meshes that do not fit the interface. Numerical evidence shows that the discrete pressure approximation may be unstable in the neighborhood of the interface, even though the spatial approximation is based on inf-sup stable finite elements. It means that XFEM enrichment locally violates the satisfaction of the stability condition for mixed problems. For this reason, resorting to pressure stabilization techniques in the region of elements cut by the unfitted interface is mandatory. In alternative, we consider the application of stabilized equal order pressure/velocity XFEM discretizations and we analyze their approximation properties. On one side, this strategy increases the flexibility on the choice of velocity and pressure approximation spaces. On the other side, symmetric pressure stabilization operators, such as local pressure projection methods or the Brezzi–Pitkaranta scheme, seem to be effective to cure the additional source of instability arising from the XFEM approximation. We will show that these operators can be applied either locally, namely only in proximity of the interface, or globally, that is on the whole domain when combined with equal order approximations. After analyzing the stability, approximation properties and the conditioning of the scheme, numerical results on benchmark cases will be discussed, in order to thoroughly compare the performance of different variants of the method.     

Process monitoring in stamping operations through tooling integrated sensing

This paper introduces a new sensing method for stamping process monitoring based on the measurement of contact pressure distribution across the sheet metal-tooling interface, by means of an array of tooling-integrated force sensors. The role of numeric surface methods in estimating the contact pressure distribution on the sheet metal-tooling interface has been studied through finite element analysis and experiments. A finite element model is set up to model the contact interactions, based on the geometry of a customized stamping test fixture. Discrete samples of contact pressure taken from the FE model have been used to recreate continuous-pressure surfaces based on the Thin Plate Splines (TPS) and Bezier surface algorithms. It is shown that the temporal–spatial contact pressure distribution across the sheet metal-tooling interface can be effectively reconstructed through interpolation using spatially discrete sensor data. Comparison of surface-based pressure estimates with the FEA field solution indicates that the TPS-based method is more accurate than the Bezier method. The effectiveness of the surface modeling scheme is also evaluated experimentally by comparing the net press force calculated from numerical integration of the TPS surfaces with the experimentally measured value under different press speeds. The effectiveness of the new sensing method is further demonstrated by detection of slide misparallelism through analysis of the tooling interface pressure distribution. The study presents a new approach to enhancing process observability in manufacturing operations.     

Large eddy simulation of turbulent heat transport in the Strait of Gibraltar

We develop a numerical model for large eddy simulation of turbulent heat transport in the Strait of Gibraltar. The flow equations are the incompressible Navier–Stokes equations including Coriolis forces and density variation through the Boussinesq approximation. The turbulence effects are incorporated in the system by considering the Smagorinsky model. As a numerical solver we propose a finite element semi-Lagrangian method. The solution procedure consists of combining a non-oscillatory semi-Lagrangian scheme for time discretization with the finite element method for space discretization. Numerical results illustrate a buoyancy-driven circulations along the Strait of Gibraltar and the sea-surface temperature is flushed out and move to northeast coast. The Ocean discharge and the temperature difference are shown to control the plume structure.     

Large eddy simulation of turbulent heat transport in the Strait of Gibraltar

We develop a numerical model for large eddy simulation of turbulent heat transport in the Strait of Gibraltar. The flow equations are the incompressible Navier–Stokes equations including Coriolis forces and density variation through the Boussinesq approximation. The turbulence effects are incorporated in the system by considering the Smagorinsky model. As a numerical solver we propose a finite element semi-Lagrangian method. The solution procedure consists of combining a non-oscillatory semi-Lagrangian scheme for time discretization with the finite element method for space discretization. Numerical results illustrate a buoyancy-driven circulations along the Strait of Gibraltar and the sea-surface temperature is flushed out and move to northeast coast. The Ocean discharge and the temperature difference are shown to control the plume structure.     

Simulation of incompressible two-phase flows with large density differences employing lattice Boltzmann and level set methods

A hybrid lattice Boltzmann and level set method (LBLSM) for two-phase immiscible fluids with large density differences is proposed. The lattice Boltzmann method is used for calculating the velocities, the interface is captured by the level set function and the surface tension force is replaced by an equivalent force field. The method can be applied to simulate two-phase fluid flows with the density ratio up to 1000. In case of zero or known pressure gradient the method is completely explicit. In order to validate the method, several examples are solved and the results are in agreement with analytical or experimental results.     

An anisotropic pressure-stabilized finite element method for incompressible flow problems

We consider a pressure-stabilized, finite element approximation of incompressible flow problems in primitive velocity–pressure variables, which is based on a projection of the gradient of the discrete pressure onto the space of discrete functions. Equal order interpolation for the velocity and the pressure can be employed with this formulation. The method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios.     

Explicit level set and density methods for topology optimization with equivalent minimum length scale constraints

The goal of this paper is to introduce local length scale control in an explicit level set method for topology optimization. The level set function is parametrized explicitly by filtering a set of nodal optimization variables. The extended finite element method (XFEM) is used to represent the non-conforming material interface on a fixed mesh of the design domain. In this framework, a minimum length scale is imposed by adopting geometric constraints that have been recently proposed for density-based topology optimization with projections filters. Besides providing local length scale control, the advantages of the modified constraints are twofold. First, the constraints provide a computationally inexpensive solution for the instabilities which often appear in level set XFEM topology optimization. Second, utilizing the same geometric constraints in both the density-based topology optimization and the level set optimization enables to perform a more unbiased comparison between both methods. These different features are illustrated in a number of well-known benchmark problems for topology optimization.

     

Surface Smoothing Procedures in Computational Contact Mechanics

This work addresses the problems arising in the finite element simulation of contact problems undergoing large deformation. The frictional contact problem is formulated in the continuum framework, introducing the interface laws for the normal and tangential stress components in the contact area. The variational formulation is presented, considering different methods to enforce the contact constraints. The spatial discretization within the finite element method is applied, as well as the temporal discretization required to solve the three sources of nonlinearities: geometric, material and frictional contact. The discretization of contact surfaces is discussed in detail, including different surface smoothing procedures. This numerical strategy allows to solve the difficulties associated with the discontinuities in the contact surface geometry introduced by finite element discretization, which leads to nonphysical oscillations of the contact force for large sliding problems. The geometrical accuracy of different interpolation methods is evaluated, paying particular attention to the Nagata patch interpolation recently proposed. In this framework, the Node-to-Nagata contact elements are developed using the augmented Lagrangian method to regularize the variational frictional contact problem. The techniques used to search for contact in case of large deformations are discussed, including self-contact phenomena. Several numerical examples are presented, comprising both the contact between deformable and rigid obstacles and the contact between deformable bodies. The results show that the accuracy and robustness of the numerical simulations is improved when the contact surface is smoothed with Nagata patches.     

Flux-based level-set method for two-phase flows on unstructured grids

In this paper we deal with the application of the flux-based level set method to moving interface computations on unstructured grids. The focus lies on the overcoming of the known difficulties of level set methods, e.g. accurate computations of important geometric properties, reliable and precise reinitialization of the level set function and the adaption of standard discretization methods to the moving boundary case. The basic building block of our approach is the high-resolution flux-based level set method for general advection equation (Frolkovi? and Mikula in SIAM J Sci Comput 29(2):579–597, 2007, Frolkovi? and Wehner in Comput Vis Sci 12(6):626–650, 2009). We extend this method for the problem of reinitialization of the level set function on unstructured grids by using quadratic interpolation to compute distances for nodes close to the interface. To realize numerical simulation for some applications with moving boundaries, we adapt the approach of ghost fluid method (Gibou and Fedkiw in J Comput Phys 202:577–601, 2005) for unstructured grids. The idea is to describe the development of the moving boundary with a level set formulation while the computational grid remains fixed and the boundary conditions are enforced using some extrapolation. Our main motivation is the numerical solution of two-phase incompressible flow problems. Additionally to previously mentioned steps, we introduce further numerical schemes in the framework of finite volume discretization for the flow. Possible jumps of the pressure and the directional derivative of velocity at the interface are modeled directly within the method using the approach of extended approximation spaces. Besides that, an algorithm for the computations of curvature is considered that exhibits the second order accuracy for some examples. Numerical experiments are provided for the presented methods.     

A numerical method for the simulation of free surface flows with surface tension

A numerical model for the simulation of three-dimensional liquid–gas flows with free surfaces and surface tension is presented. The incompressible Navier–Stokes equations are assumed to hold in the liquid domain, while the gas pressure is assumed to be constant in each connected component of the gas domain and to follow the ideal gas law. The surface tension effects are imposed as a normal force on the interface.

An implicit splitting scheme is used to decouple the physical phenomena. Given the curvature of the liquid–gas interface, the method described in [Caboussat A, Picasso M, Rappaz J. Numerical simulation of free surface incompressible liquid flows surrounded by compressible gas. J Comput Phys 2005;203(2):626–49] is used to track the liquid domain and compute the velocity and pressure in the liquid and the pressure in the gas domain. Then the surface tension effects are added. A variational method for the computation of the curvature is presented by smoothing the characteristic function of the liquid domain and using a finite element unstructured mesh.

The model is validated and numerical results in two and three space dimensions are presented for bubbles and/or droplets flows.     

Levelset based fluid topology optimization using the extended finite element method

This study focuses on finding the optimal layout of fluidic devices subjected to incompressible flow at low Reynolds numbers. The proposed approach uses a levelset method to describe the fluid-solid interface geometry. The flow field is modeled by the incompressible Navier?CStokes equations and discretized by the extended finite element method (XFEM). The no-slip condition along the fluid-solid interface is enforced via a stabilized Lagrange multiplier method. Unlike the commonly used porosity approach, the XFEM approach does not rely on a material interpolation scheme, which allows for more flexibility in formulating the design problems. Further, it mitigates shortcomings of the porosity approach, including spurious pressure diffusion through solid material, strong dependency of the accuracy of the boundary enforcement with respect to the model parameters which may affect the optimization results, and poor boundary resolution. Numerical studies verify that the proposed method is able to recover optimization results obtained with the porosity approach. Further, it is demonstrated that the XFEM approach yields physical results for problems that cannot be solved with the porosity approach.     

Coupling finite elements and auxiliary sources

We consider second-order scalar elliptic boundary value problems on unbounded domains, which model, for instance, electrostatic fields. We propose a discretization that relies on a Trefftz approximation by multipole auxiliary sources in some parts of the domain and on standard mesh-based primal Lagrangian finite elements in other parts. Several approaches are developed and, based on variational saddle point theory, rigorously analyzed to couple both discretizations across the common interface:1. Least-squares-based coupling using techniques from PDE-constrained optimization.2. Coupling through Dirichlet-to-Neumann operators.3. Three-field variational formulation in the spirit of mortar finite element methods.We compare these approaches in a series of numerical experiments.     

A Continuum Mechanical Approach to Geodesics in Shape Space

In this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes, where shapes are implicitly described as boundary contours of objects. The proposed shape metric is derived from a continuum mechanical notion of viscous dissipation. A geodesic path is defined as the family of shapes such that the total amount of viscous dissipation caused by an optimal material transport along the path is minimized. The approach can easily be generalized to shapes given as segment contours of multi-labeled images and to geodesic paths between partially occluded objects. The proposed computational framework for finding such a minimizer is based on the time discretization of a geodesic path as a sequence of pairwise matching problems, which is strictly invariant with respect to rigid body motions and ensures a 1–1 correspondence along the induced flow in shape space. When decreasing the time step size, the proposed model leads to the minimization of the actual geodesic length, where the Hessian of the pairwise matching energy reflects the chosen Riemannian metric on the underlying shape space. If the constraint of pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional, one obtains for decreasing time step size an optical flow term controlling the transport of the shape by the underlying motion field. The method is implemented via a level set representation of shapes, and a finite element approximation is employed as spatial discretization both for the pairwise matching deformations and for the level set representations. The numerical relaxation of the energy is performed via an efficient multi-scale procedure in space and time. Various examples for 2D and 3D shapes underline the effectiveness and robustness of the proposed approach.     

A simple stabilized finite element method for solving two phase compressible–incompressible interface flows

When computing interface flows between compressible (gas) and incompressible (liquid) fluids, one faces at least to the following difficulties: (1) transition from a gas density linked to the local temperature and pressure by an equation of state to a liquid density mainly constant in space, (2) proper approximation of the divergence constraint in incompressible regions and (3) wave transmission at the interface. The aim of the present paper is to design a global (i.e. the same for each phase) numerical method to address easily this coupling. To this end, the same set of primitive unknowns and equations is used everywhere in the flow, but with a dynamic parameterization that changes from compressible to incompressible regions. On one hand, the compressible Navier–Stokes equations are considered under weakly compressibility assumption so that a non-conservative formulation can be used. On the other hand, the incompressible non-isothermal model is retained. In addition, the level set transport equation is used to capture the interface position needed to identify the local characteristics of the fluid and to recover the adequate local modelling. For space approximation of Navier–Stokes equations, a Galerkin least-squares finite element method is used. Two essential elements for defining this numerical scheme are the stabilization and the computation of element integral of the approximated weak form. Since very different concerns motivate the need for stabilization in compressible and incompressible flows, the first difficulty is to design a stabilization operator suitable for both types of flows especially in mixed elements. In addition, some integral of discontinuous functions must be correctly computed to ensure interfacial wave transmission. To overcome these two difficulties, specific averages are computed especially near the interface. Finally, the level set transport equation is computed by a quadrature free Discontinuous Galerkin method. Numerical strategies are performed and validated for 1D and 2D applications.     

Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem

In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by semilinear parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. By applying some error estimates techniques of mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present a numerical example which confirms our theoretical results.     

On error estimates of the fully discrete penalty method for the viscoelastic flow problem

In this paper, a fully discrete finite element penalty method is presented for the two-dimensional viscoelastic flow problem arising in the Oldroyd model, in which the spatial discretization is based on the finite element approximation and the time discretization is based on the backward Euler scheme. Moreover, we provide the optimal error estimate for the numerical solution under some realistic assumptions. Finally, some numerical experiments are shown to illustrate the efficiency of the penalty method.