A stabilized incremental projection scheme for the incompressible Navier–Stokes equations

It is well known that any spatial discretization of the saddle‐point Stokes problem should satisfy the Ladyzhenskaya–Brezzi–Babuska (LBB) stability condition in order to prevent the appearance of spurious pressure modes. Particularly, if an equal‐order approximation is applied, the Schur complement (or, as called some times, the Uzawa matrix) of the pressure system has a non‐trivial null space that gives rise to such modes. An idea in the past was that all the schemes that solve a Poisson equation for the pressure rather than the Uzawa pressure equation (splitting/projection methods) should overcome this difficulty; this idea was wrong. There is numerical evidence that at least the so‐called incremental projection scheme still suffers from spurious pressure oscillations if an equal‐order approximation is applied. The present paper tries to distinguish which projection requires LBB‐compliant approximation and which does not. Moreover, a stabilized version of the incremental projection scheme is derived. Proper bounds for the stabilization parameter are also given. The numerical results show that the stabilized scheme does indeed achieve second‐order accuracy and does not produce spurious (node to node) pressure oscillations. Copyright © 2001 John Wiley & Sons, Ltd.     

Two-level stabilized finite element method for Stokes eigenvalue problem

A two-level stabilized finite element method for the Stokes eigenvalue problem based on the local Gauss integration is considered.This method involves solving a Stokes eigenvalue problem on a coarse mesh with mesh size H and a Stokes problem on a fine mesh with mesh size h = O(H 2),which can still maintain the asymptotically optimal accuracy.It provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution,which involves solving a Stokes eigenvalue problem on a fine mesh with mesh size h.Hence,the two-level stabilized finite element method can save a large amount of computational time.Moreover,numerical tests confirm the theoretical results of the present method.     

一种新的求解对流占优问题的自适应网格细化方法

在均匀网格上求解对流占优问题时,往往会产生数值震荡现象,因此需要局部加密网格来提高解的精度。针对对流占优问题,设计了一种新的自适应网格细化算法。该方法采用流线迎风SUPG(Petrov-Galerkin)格式求解对流占优问题,定义了网格尺寸并通过后验误差估计子修正来指导自适应网格细化,以泡泡型局部网格生成算法BLMG为网格生成器,通过模拟泡泡在区域中的运动得到了高质量的点集。与其他自适应网格细化方法相比,该方法可在同一框架内实现网格的细化和粗化,同时在所有细化层得到了高质量的网格。数值算例结果表明,该方法在求解对流占优问题时具有更高的数值精度和更好的收敛性。     

An efficient implicit mesh‐less method for compressible flow calculations

A dual‐time implicit mesh‐less scheme is presented for calculation of compressible inviscid flow equations. The Taylor series least‐square method is used for approximation of spatial derivatives at each node which leads to a central difference discretization. Several convergence acceleration techniques such as local time stepping, enthalpy damping and residual smoothing are adopted in this approach. The capabilities of the method are demonstrated by flow computations around single and multi‐element airfoils at subsonic, transonic and supersonic flow conditions. Results are presented which indicate good agreements with other reliable finite‐volume results. The computational time is considerably reduced when using the proposed mesh‐less method compared with the explicit mesh‐less and finite‐volume schemes using the same point distributions. Copyright © 2010 John Wiley & Sons, Ltd.     

Hybrid projection method for generalized mixed equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces

A new hybrid projection iterative scheme is introduced to approximate a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of fixed points of a relatively weak nonexpansive mapping in the Banach spaces. The obtained results generalize and improve the recent results announced by many other authors.     

Hybrid projection method for generalized mixed equilibrium problems,variational inequality problems,and fixed point problems in Banach spaces

A new hybrid projection iterative scheme is introduced to approximate a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of fixed points of a relatively weak nonexpansive mapping in the Banach spaces. The obtained results generalize and improve the recent results announced by many other authors.     

An unfitted interior penalty discontinuous Galerkin method for incompressible Navier–Stokes two‐phase flow

A discontinuous Galerkin method for the solution of the immiscible and incompressible two‐phase flow problem based on the nonsymmetric interior penalty method is presented. Therefore, the incompressible Navier–Stokes equation is solved for a domain decomposed into two subdomains with different values of viscosity and density as well as a singular surface tension force. On the basis of a piecewise linear approximation of the interface, meshes for both phases are cut out of a structured mesh. The discontinuous finite elements are defined on the resulting Cartesian cut‐cell mesh and may therefore approximate the discontinuities of the pressure and the velocity derivatives across the interface with high accuracy. As the mesh resolves the interface, regularization of the density and viscosity jumps across the interface is not required. This preserves the local conservation property of the velocity field even in the vicinity of the interface and constitutes a significant advantage compared with standard methods that require regularization of these discontinuities and cannot represent the jumps and kinks in pressure and velocity. A powerful subtessellation algorithm is incorporated to allow the usage of standard time integrators (such as Crank–Nicholson) on the time‐dependent mesh. The presented discretization is applicable to both the two‐dimensional and three‐dimensional cases. The performance of our approach is demonstrated by application to a two‐dimensional benchmark problem, allowing for a thorough comparison with other numerical methods. Copyright © 2012 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.     

Anisotropic goal‐oriented error analysis for a third‐order accurate CENO Euler discretization

In this paper, a central essentially non‐oscillatory approximation based on a quadratic polynomial reconstruction is considered for solving the unsteady 2D Euler equations. The scheme is third‐order accurate on irregular unstructured meshes. The paper concentrates on a method for a metric‐based goal‐oriented mesh adaptation. For this purpose, an a priori error analysis for this central essentially non‐oscillatory scheme is proposed. It allows us to get an estimate depending on the polynomial reconstruction error. As a third‐order error is not naturally expressed in terms of a metric, we propose a least‐square method to approach a third‐order error by a quadratic term. Then an optimization problem for the best mesh metric is obtained and analytically solved. The resulting mesh optimality system is discretized and solved using a global unsteady fixed‐point algorithm. The method is applied to an acoustic propagation benchmark.     

A preconditioned conjugate gradient Uzawa-type method for the solution of the stokes problem by mixed Q1–P0 stabilized finite elements

We study the behaviour of a conjugate gradient Uzawa-type method for a stabilized finite element approximation of the Stokes problem. Many variants of the Uzawa algorithm have been described for different finite elements satisfying the well-known Inf-Sup condition of Babu?ka and Brezzi, but it is surprising that developments for unstable ‘low-order’ discretizations with stabilization procedures are still missing. Our paper is presented in this context for the popular (so-called) Q1–P0 element. First we show that a simple stabilization technique for this element permits us to retain the property of a convergence factor bounded independently of the discretization mesh size. The second contribution of this work deals with the construction of a less costly preconditioner taking full advantages of the block diagonal structure of the stabilization matrix. Its efficiency is supported by 2D and. 3D numerical results.     

FREE BOUNDARY PROBLEM OF THE 2D SEEPAGE FLOW

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Nontraditional Substructuring Procedure for Analysis of Free-Form Shells

An efficient nontraditional scheme of the substructuring method is expounded. It is based on the curvilinear mesh method (CMM) used for analysis of complex shell structures. In forming the stiffness matrix, this approach excludes fully the shell displacement approximation errors. A numerical algorithm that preserves the efficiency of the CMM for shells of arbitrary form is developed. The stress–strain problem for a boxed beam with diaphragms and a cylindrical roof shell is solved numerically. The results are compared with those obtained by other numerical methods     

Stability of algebraic multigrid for Stokes problems

An investigation is made of the performance of algebraic multigrid (AMG) solvers for the discrete Stokes problem. The saddle‐point formulations are based on the direct enforcement of the fundamental conservation laws in discrete spaces and subsequently stabilised with the aid of a regular splitting of the diffusion operator. AMG solvers based on an independent coarsening of the fields (the unknown approach) and also on a common coarsening (the point approach) are investigated. Both mixed‐order and equal‐order interpolations are considered. The dependence of convergence on the ‘degree of coarsening’ is investigated by studying the ‘convergence versus coarsening’ characteristics and their variation with mesh resolution. They show a consistency in shape, which reveals two distinct performance zones, one convergent the other divergent. The transition from the convergent to the divergent zones is discontinuous and occurs at a critical coarsening factor that is largely mesh independent. It signals a breakdown in the stability of the smoothing at the coarser levels of coarse grid approximation. It is shown that the previously observed, mesh‐dependent, scaling of convergence factors, which had suggested inconsistencies in the coarse grid approximation, is not a reliable marker of inconsistency. It is an indirect consequence of the breakdown in the stability of smoothing. For stable smoothing, reduction factors are shown to be largely mesh independent. The ability of mixed‐order interpolation to permit stable smoothing and therefore to deliver mesh‐independent convergence is explained. Two expedient options are suggested for obtaining mesh‐independent convergence for those AMG codes that are based on an equal‐order interpolation. Copyright © 2012 John Wiley & Sons, Ltd.     

A stabilized finite element formulation for high‐speed inviscid compressible flows using finite calculus

This paper aims at the development of a new stabilization formulation based on the finite calculus (FIC) scheme for solving the Euler equations using the Galerkin FEM on unstructured triangular grids. The FIC method is based on expressing the balance of fluxes in a space–time domain of finite size. It is used to prevent the creation of instabilities typically present in numerical solutions due to the high convective terms and sharp gradients. Two stabilization terms, respectively called streamline term and transverse term, are added via the FIC formulation to the original conservative equations in the space–time domain. An explicit fourth‐order Runge–Kutta scheme is implemented to advance the solution in time. The presented numerical test examples for inviscid flows prove the ability of the proposed stabilization technique for providing appropriate solutions especially near shock waves. Although the derived methodology delivers precise results with a nearly coarse mesh, a mesh refinement technique is coupled to the solution process for obtaining a suitable mesh particularly in the high‐gradient zones. Copyright © 2014 John Wiley & Sons, Ltd.     

An unstructured finite element model for incompressible two-phase flow based on a monolithic conservative level set method

We present a robust numerical method for solving incompressible, immiscible two-phase flows. The method extends both a monolithic phase conservative level set method with embedded redistancing and a semi-implicit high-order projection scheme for variable-density flows. The level set method can be initialized conveniently via a simple phase indicator field instead of a signed distance function (SDF). To process the indicator field into a SDF, we propose a new partial differential equation-based redistancing method. We also improve the monolithic level set scheme to provide more accuracy and robustness in full two-phase flow simulations. Specifically, we perform an extra step to ensure convergence to the signed distance level set function and simplify other aspects of the original scheme. Lastly, we introduce consistent artificial viscosity to stabilize the momentum equations in the context of the projection scheme. This stabilization is algebraic, has no tunable parameters and is suitable for unstructured meshes and arbitrary refinement levels. The overall methodology includes few numerical tuning parameters; however, for the wide range of problems that we solve, we identify only one parameter that strongly affects performance of the computational model and provide a value that provides accurate results across all the benchmarks presented. This methodology results in a robust, accurate, and efficient two-phase flow model, which is mass- and volume-conserving on unstructured meshes and has low user input requirements, making it attractive for real-world applications.     

A posteriori bounds for linear functional outputs of Crouzeix–Raviart finite element discretizations of the incompressible Stokes problem

A finite element technique is presented for the efficient generation of lower and upper bounds to outputs which are linear functionals of the solutions to the incompressible Stokes equations in two space dimensions. The finite element discretization is effected by Crouzeix–Raviart elements, the discontinuous pressure approximation of which is central to this approach. The bounds are based upon the construction of an augmented Lagrangian: the objective is a quadratic ‘energy’ reformulation of the desired output, the constraints are the finite element equilibrium equations (including the incompressibility constraint), and the inter‐sub‐domain continuity conditions on velocity. Appealing to the dual max–min problem for appropriately chosen candidate Lagrange multipliers then yields inexpensive bounds for the output associated with a fine‐mesh discretization. The Lagrange multipliers are generated by exploiting an associated coarse‐mesh approximation. In addition to the requisite coarse‐mesh calculations, the bound technique requires the solution of only local sub‐domain Stokes problems on the fine mesh. The method is illustrated for the Stokes equations, in which the outputs of interest are the flow rate past and the lift force on a body immersed in a channel. Copyright © 2000 John Wiley & Sons, Ltd.     

SELF-ADAPTIVE STRATEGY FOR ONE-DIMENSIONAL FINITE ELEMENT METHOD BASED ON ELEMENT ENERGY PROJECTION METHOD

Based on the newly-developed element energy projection (EEP) method for computation of super-convergent results in one-dimensional finite element method (FEM), the task of self-adaptive FEM analysis was converted into the task of adaptive piecewise polynomial interpolation. As a result, a satisfactory FEM mesh can be obtained, and further FEM analysis on this mesh would immediately produce an FEM solution which usually satisfies the user specified error tolerance. Even though the error tolerance was not completely satisfied, one or two steps of further local refinements would be sufficient. This strategy was found to be very simple, rapid, cheap and efficient. Taking the elliptical ordinary differential equation of second order as the model problem, the fundamental idea, implementation strategy and detailed algorithm are described. Representative numerical examples are given to show the effectiveness and reliability of the proposed approach.     

A multiscale control volume finite element method for advection–diffusion equations

We present a new stabilized method for advection–diffusion equations, which combines a control volume FEM formulation of the governing equations with a novel multiscale approximation of the total flux. The latter incorporates information about the exact solution that cannot be represented on the mesh. To define this flux, we solve the governing equations along suitable mesh segments under the assumption that the flux varies linearly along these segments. This procedure yields second‐order accurate fluxes on the edges of the mesh. Then, we use curl‐conforming elements of the same order to lift these edge fluxes into the mesh elements. In so doing, we obtain a stabilized control volume FEM formulation that is second‐order accurate and does not require mesh‐dependent stabilization parameters. Numerical convergence studies on uniform and nonuniform grids along with several standard advection tests illustrate the computational properties of the new method. Published 2015. This article is a U.S. Government work and is in the public domain in the USA.     

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

In 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.     

Adaptive mesh refinement for simulation of thin film flows

We present a robust and accurate numerical method for simulating gravity-driven, thin-film flow problems. The convection term in the governing equation is treated by a semi-implicit, essentially non-oscillatory scheme. The resulting nonlinear discrete equation is solved using a nonlinear full approximation storage multigrid algorithm with adaptive mesh refinement techniques. A set of representative numerical experiments are presented. We show that the use of adaptive mesh refinement reduces computational time and memory compared to the equivalent uniform mesh results. Our simulation results are consistent with previous experimental observations.