A simple finite element method for Stokes flows with surface tension using unfitted meshes

This paper presents a simple finite element method for Stokes flows with surface tension. The method uses an unfitted mesh that is independent of the interface. Due to the surface force, the pressure has a jump across the interface. Based on the properties of the level set function that implicitly represents the interface, the jump of the pressure is removed, and a new problem without discontinuities is formulated. Then, classical stable finite element methods are applied to solve the new problem. Some techniques are used to show that the method is equivalent to an easy‐to‐implement method that can be regarded as a traditional method with a modified pressure space. However, the matrix of the resulting linear system of equations is the same as that of the traditional method. Optimal error estimates are derived for the proposed method. Finally, some numerical tests are presented to confirm the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

A finite element method for analysis of fluid flow,heat transfer and free interfaces in Czochralski crystal growth

A finite element algorithm is presented for simultaneous calculation of the steady state, axisymmetric flows and the crystal, melt/crystal and melt/ambient interface shapes in the Czochralski technique for crystal growth from the melt. The analysis is based on mixed Lagrangian finite element approximations to the velocity, temperature and pressure fields and isoparametric approximations to the interface shape. Galerkin's method is used to reduce the problem to a non-linear algebraic set, which is solved by Newton's method. Sample solutions are reported for the thermophysical properties appropriate for silicon, a low-Prandtl-number semiconductor, and for GGG, a high–Prandtl–number oxide material. The algorithm is capable of computing solutions for both materials at realistic values of the Grashof number, and the calculations are convergent with mesh refinement. Flow transitions and interface shapes are calculated as a function of increasing flow intensity and compared for the two material systems. The flow pattern near the melt/gas/crystal tri-junction has the asymptotic form predicted by an inertialess analysis assuming the meniscus and solidification interfaces are fixed.     

Comparison of spectral and finite element methods applied to the study of the core‐annular flow in an undulating tube

A Galerkin/finite element and a pseudo‐spectral method, in conjunction with the primitive (velocity‐pressure) and streamfunction‐vorticity formulations, are tested for solving the two‐phase flow in a tube, which has a periodically varying, circular cross section. Two immiscible, incompressible, Newtonian fluids are arranged so that one of them is around the axis of the tube (core fluid) and the other one surrounds it (annular fluid). The physical and flow parameters are such that the interface between the two fluids remains continuous and single‐valued. This arrangement is usually referred to as Core‐Annular flow. A non‐orthogonal mapping is used to transform the uneven tube shape and the unknown, time dependent interface to fixed, cylindrical surfaces. With both methods and formulations, steady states are calculated first using the Newton–Raphson method. The most dangerous eigenvalues of the related linear stability problem are calculated using the Arnoldi method, and dynamic simulations are carried out using the implicit Euler method. It is shown that with a smooth tube shape the pseudo‐spectral method exhibits exponential convergence, whereas the finite element method exhibits algebraic convergence, albeit of higher order than expected from the relevant theory. Thus the former method, especially when coupled with the streamfunction‐vorticity formulation, is much more efficient. The finite element method becomes more advantageous when the tube shape contains a cusp, in which case the convergence rate of the pseudo‐spectral method deteriorates exhibiting algebraic convergence with the number of the axial spectral modes, whereas the convergence rate of the finite element method remains unaffected. Copyright © 2002 John Wiley & Sons, Ltd.     

自适应分析在确定裂纹尖端塑性区中的应用

在分析裂纹扩展及材料强度时对塑性区的估算是很重要的,本文提出采用自适应有限元分析来确定裂纹尖端塑性区的方法,计算结果表明这种方法通过网格自动加密能够有效地跟踪出弹塑性的交界面。     

Shape optimization of a body located in incompressible flow using adjoint method and partial control algorithm

The purpose of this study is to obtain an optimal shape of a body located in an incompressible viscous flow. The optimal shape of the body is defined so as to minimize the fluid forces acting on it by determining the surface coordinates based on the finite element method and the optimal control theory. The performance function, which is used to judge the optimality of a shape, is defined as the square sum of the drag and lift forces. The minimization problem is solved using an adjoint equation method. The gradient in the adjoint equation is affected by the finite element configuration. The use of a finite element mesh whose shape is appropriate for the procedure is important in shape optimization. If the finite element mesh used is not suitable for computations, the exact gradient is not calculated. Therefore, a structured mesh is used for the adjacent area of the body and all finite element meshes are refined using the Delaunay triangulation at each iteration computation. The weighted gradient method is applied as the minimization technique. Using an algorithm in which all nodal coordinates on the surface of the body are employed and starting from a circle as an initial shape, a front‐edged and rear‐round shape is obtained because of the vortices at the back of the body. To overcome this difficulty, we introduced the partial control algorithm, in which some of the nodal coordinates on the surface of the body are updated. From four cases of computational studies, we reveal that the optimal shape has both sharp front and sharp rear edges. All computations are conducted at Reynolds number Re=250. The minimum value of the performance function is obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

Spurious currents in finite element based level set methods for two‐phase flow

A study of spurious currents in continuous finite element based simulations of the incompressible Navier–Stokes equations for two‐phase flows is presented on the basis of computations on a circular drop in equilibrium. The conservative and the standard level set methods are used. It is shown that a sharp surface tension force, expressed as a line integral along the interface, can give rise to large spurious currents and oscillations in the pressure that do not decrease with mesh refinement. If instead a regularized surface tension representation is used, exact force balance at the interface is possible, both for a fully coupled discretization approach and for a fractional step projection method. However, the numerical curvature calculation introduces errors that cause spurious currents. Different ways to extend the curvature from the interface to the whole domain are discussed and investigated. The impact of using different finite element spaces and stabilization methods is also considered. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical investigations of the XFEM for solving two-phase incompressible flows

This paper discusses the application of the extended finite element method (XFEM) to solve two-phase incompressible flows. The Navier–Stokes equations are discretised using the Taylor–Hood finite element. To capture the different discontinuities across the interface, kink or jump enrichments are used for the velocity and/or pressure fields. However, these enrichments may lead to an inappropriate combination of interpolations. Different polynomial enrichment orders and different enrichment functions are investigated; only the stable combination will be used afterward.

In cases with a surface tension force, the accuracy mainly relies on the precise computation of the normal and curvature. A novel method for computing normal vectors to the interface is proposed. This method employs successive mesh refinements inside the cut elements. Comparisons with analytical and numerical solutions demonstrate that the method is effective. Moreover, the mesh refinement improves the sub-integration in the XFEM and allows for a precise re-initialisation procedure.     

Velocity–pressure integrated versus penalty finite element methods for high-Reynolds-number flows

Velocity–pressure integrated and consistent penalty finite element computations of high-Reynolds-number laminar flows are presented. In both methods the pressure has been interpolated using linear shape functions for a triangular element which is contained inside the biquadratic flow element. It has been shown previously that the pressure interpolation method, when used in conjunction with the velocity-pressure integrated method, yields accurate computational results for high-Reynolds-number flows. It is shown in this paper that use of the same pressure interpolation method in the consistent penalty finite element method yields computational results which are comparable to those of the velocity–pressure integrated method for both the velocity and the pressure fields. Accuracy of the two finite element methods has been demonstrated by comparing the computational results with available experimental data and/or fine grid finite difference computational results. Advantages and disadvantages of the two finite element methods are discussed on the basis of accuracy and convergence nature. Example problems considered include a lid-driven cavity flow of Reynolds number 10 000, a laminar backward-facing step flow and a laminar flow through a nest of cylinders.     

A control volume finite element method for three-dimensional three-phase flows

A novel control volume finite element method with adaptive anisotropic unstructured meshes is presented for three-dimensional three-phase flows with interfacial tension. The numerical framework consists of a mixed control volume and finite element formulation with a new P₁DG-P₂ elements (linear discontinuous velocity between elements and quadratic continuous pressure between elements). A “volume of fluid” type method is used for the interface capturing, which is based on compressive control volume advection and second-order finite element methods. A force-balanced continuum surface force model is employed for the interfacial tension on unstructured meshes. The interfacial tension coefficient decomposition method is also used to deal with interfacial tension pairings between different phases. Numerical examples of benchmark tests and the dynamics of three-dimensional three-phase rising bubble, and droplet impact are presented. The results are compared with the analytical solutions and previously published experimental data, demonstrating the capability of the present method.     

多尺度六面体单元网格连接方法研究

提出多结点六面体单元的结点形函数构建方法,解决了数字化分析时两个多尺度有限元网格在接触界面由于单元结点不匹配而导致的结点属性不能连续传递问题.首先将全局坐标下的六面体单元及其表面上多结点通过等参逆变换转成局部坐标下的规则六面体单元及其表面上多结点;在规则单元中,以每个结点为基点,分别沿三个正交的局部坐标方向在单元内寻找...     

节点梯度光滑有限元配点法

配点法构造简单、计算高效,但需要用到数值离散形函数的高阶梯度,而传统有限元形函数的梯度在单元边界处通常仅具有C0连续性,因此无法直接用于配点法分析.本文通过引入有限元形函数的光滑梯度,提出了节点梯度光滑有限元配点法.首先基于广义梯度光滑方法,定义了有限元形函数在节点处的一阶光滑梯度值,然后以有限元形函数为核函数构造了有...     

Solution of flow around complex‐shaped surfaces by an immersed boundary‐body conformal enrichment method

Solving the flow around objects with complex shapes may involve extensive meshing work that has to be repeated each time a change in the geometry is needed. Time consuming meshing can be avoided when the solution algorithm can tackle grids that do not fit the shape of immersed objects. This work presents applications of a recently proposed immersed boundary—body conformal enrichment method to the solution of the flow around complex shaped surfaces such as those of a metallic foam matrix. The method produces solutions of the flow satisfying accurately Dirichlet boundary conditions imposed on the immersed fluid/solid interface. The boundary of immersed objects is defined using a level‐set function, and the finite element discretization of interface elements is enriched with additional degrees of freedom, which are eliminated at element level. The method is first validated in the case of flow problems for which reference solutions on body‐conformal grids can be obtained: flow around an array of spheres and flow around periodic arrays of cylinders. Then, solutions are shown for the more complex flow inside a metallic foam matrix. A multiscale approach combining the solution at the pore level by the immersed boundary method and the macro‐scale solution with simulated permeability is used to solve actual experimental configurations. The computed pressure drop as a function of the flow rate on the macro scale configuration replicating two experimental setups is compared with the experimental data for various foam thicknesses. Copyright © 2011 National Research Council Canada     

多边形有限元研究进展

有限元法是数值求解偏微分方程边值问题的重要方法,采用不规则多边形单元网格, 可以方便有效地模拟材料的力学性能, 又使得区域网格剖分变得灵活方便. 特别是对于复杂的几何形状, 多边形单元网格具有更大的优势. 本文对国内外有关多边形有限元法的最新进展作了初步的总结和评述, 主要以基于位移法的多边形有限元为主.论述了多边形有限元的发展历史, 给出了多边形单元上的Wachspress插值、Laplace插值和重心坐标的一些最新研究成果. 与经典有限元法形函数为多项式形式不同, 多边形单元的形函数为有理函数或者无理函数形式. 多边形单元插值形函数满足线性完备性, 可以再现线性位移场, 像经典有限元法一样直接施加本质边界条件; 插值函数在多边形的边界上是线性的,确保不同单元间的自动协调. 不同单元的插值形函数表达公式形式统一, 方便混合单元网格计算的程序编写. 提出了多边形有限元法今后需要研究的问题.      

A singular finite element for Stokes flow: The stick–slip problem

Abrupt changes in boundary conditions in viscous flow problems give rise to stress singularities. Ordinary finite element methods account effectively for the global solution but perform poorly near the singularity. In this paper we develop singular finite elements, similar in principle to the crack tip elements used in fracture mechanics, to improve the solution accuracy in the vicinity of the singular point and to speed up the rate of convergence. These special elements surround the singular point, and the corresponding field shape functions embody the form of the singularity. Because the pressure is singular, there is no pressure node at the singular point. The method performs well when applied to the stick–slip problem and gives more accurate results than those from refined ordinary finite element meshes.     

非匹配结点的有限单元等参插值方法及其应用研究

针对两零件的异构网格单元结点在接触界面不能相互匹配导致结点属性不能连续过渡和传递的问题,提出非匹配结点的有限单元等参插值方法,通过构建非匹配结点的形函数和修正原结点的形函数,将结点属性值的影响范围限制在可控的局部区域,从而实现两异构网格结点属性在接触界面的连续过渡和传递。通过两个异构的四边形单元网格的结点属性在接触界面的过渡实例和啮合齿轮的接触分析应用,验证了该方法的正确性和有效性。     

Analysis of high-speed continuous casting with inverse finite elements

A recently proposed inverse isotherm finite element method is further extended in order to account for processes with distorted isotherms. With this method a variety of problems can be solved which require the explicit calculation of characteristic material lines along with the common field of unknowns in transport phenomena. The method is applied to high-speed metal casting, where the location and shape of the extensive solidification front is calculated simultaneously with the primary unknowns, the velocity and the pressure, whereas the temperature is fixed at the moving nodes of the finite element tessellation. This is achieved by solving the energy equation inversely along with the rest of the conservation equations, i.e. the temperature field is fixed and its location is calculated. Empirical correlations may be derived which give the shape of the solidification front as a function of the process parameters. This may be used to improve the control means of metal casting, which is currently based on one-dimensional approximate analyses.     

A hybrid scheme based on finite element/volume methods for two immiscible fluid flows

We have successfully extended our implicit hybrid finite element/volume (FE/FV) solver to flows involving two immiscible fluids. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix‐free implicit cell‐centered FV method. The pressure Poisson equation is solved by the node‐based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. This updating strategy can be rigorously proven to be able to eliminate the unphysical pressure boundary layer and is crucial for the correct temporal convergence rate. Our current staggered‐mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centers and the auxiliary variable at vertices. The fluid interface is captured by solving an advection equation for the volume fraction of one of the fluids. The same matrix‐free FV method, as the one used for momentum equations, is used to solve the advection equation. We will focus on the interface sharpening strategy to minimize the smearing of the interface over time. We have developed and implemented a global mass conservation algorithm that enforces the conservation of the mass for each fluid. Copyright © 2009 John Wiley & Sons, Ltd.     

Front Tracking Using a Hybrid Analytical Finite Element Approach for Two-Phase Flow Applied to Supercritical CO<Subscript>2</Subscript> Replacing Brine in a Heterogeneous Reservoir and Caprock

Predicting fluid replacement by two-phase flow in heterogeneous porous media is of importance for issues such as supercritical CO₂ sequestration, the integrity of caprocks and the operation of oil water/brine systems. When considering coupled process modelling, the location of the interface is of importance as most of the significant interaction between processes will be happening there. Modelling two-phase flow using grid based techniques presents a problem as the fluid–fluid interface location is approximated across the scale of the discretisation. Adaptive grid methods allow the discretisation to follow the interface through the model, but are computationally expensive and make coupling to other processes (thermal, mechanical and chemical) complicated due to the constant alteration in grid size and effects thereof. Interface tracking methods have been developed that apply sophisticated reconstruction algorithms based on either the ratio of volumes of a fluid in an element (Volume of Fluid Methods) or the advective velocity of the interface throughout the modelling regime (Level set method). In this article, we present an “Analytical Front Tracking” method where a generic analytical solution for two-phase flow is used to “add information” to a finite element model. The location of the front within individual geometrical elements is predicted using the saturation values in the elements and the velocity field of the element. This removes the necessity for grid adaptation, and reduces the need for assumptions as to the shape of the interface as this is predicted by the analytical solution. The method is verified against a standard benchmark solution and then applied to the case of CO₂ pooling and forcing its way into a heterogeneous caprock, replacing hot brine and eventually breaking through. Finally the method is applied to simulate supercritical CO₂ injected into a brine saturated heterogeneous reservoir rock leading to significant viscous fingering and developement of preferential flow paths. The results are compared with to a finite volume simulation.     

Nonlinear Parameter Identification of a Mechanical Interface Based on Primary Wave Scattering

We study stress-wave propagation in an impulsively forced split Hopkinson bar system incorporating a threaded interface. We first consider only primary transmission and reflection and reduce the problem to a first-order, strongly nonlinear ordinary differential equation governing the displacement across the interface, called the primary-pulse model. The interface is modeled as an adjusted-Iwan element, which is characterized by matching experimental and numerical eigenfrequencies as well as primary pulse amplitudes. We find that the adjusted-Iwan element parameters are dependent on preload and impact velocity (input force). A high-order finite element model paired with the identified adjusted-Iwan element is used to simulate multiple transmissions and reflections across the interface. We find that the finite element simulation reproduces the experimental results in both the wavelet and Fourier domains, validating the identification method. Our findings demonstrate that the primary-pulse model can be used for experimental parameter identification of nonlinear interfaces in waveguides.