Stability of some low‐order approximations for the Stokes problem

Two‐level low‐order finite element approximations are considered for the inhomogeneous Stokes equations. The elements introduced are attractive because of their simplicity and computational efficiency. In this paper, the stability of a Q₁(h)–Q₁(2h) approximation is analysed for general geometries. Using the macroelement technique, we prove the stability condition for both two‐ and three‐dimensional problems. As a result, optimal rates of convergence are found for the velocity and pressure approximations. Numerical results for three test problems are presented. We observe that for the computed examples, the accuracy of the two‐level bilinear approximation is compared favourably with some standard finite elements. Copyright © 2007 John Wiley & Sons, Ltd.     

An application of Discontinuous Galerkin space and velocity discretisations to the solution of a model kinetic equation

An approach based on a Discontinuous Galerkin discretisation is proposed for the Bhatnagar–Gross–Krook model kinetic equation. This approach allows for a high-order polynomial approximation of molecular velocity distribution function both in spatial and velocity variables. It is applied to model one-dimensional normal shock wave and heat transfer problems. Convergence of solutions with respect to the number of spatial cells and velocity bins is studied, with the degree of polynomial approximation ranging from zero to four in the physical space variable and from zero to eight in the velocity variable. This approach is found to conserve mass, momentum and energy when high-degree polynomial approximations are used in the velocity space. For the shock wave problem, the solution is shown to exhibit accelerated convergence with respect to the velocity variable. Convergence with respect to the spatial variable is in agreement with the order of the polynomial approximation used. For the heat transfer problem, it was observed that convergence of solutions obtained by high-degree polynomial approximations is only second order with respect to the resolution in the spatial variable. This is attributed to the temperature jump at the wall in the solutions. The shock wave and heat transfer solutions are in excellent agreement with the solutions obtained by a conservative finite volume scheme.     

A mixed galerkin method for computing the flow between eccentric rotating cylinders

A mixed Galerkin technique with B-spline basis functions is presented to compute two-dimensional incompressible flow in terms of the primitive variable formulation. To circumvent the Babuska–Brezzi stability criterion, the artificial compressibility formulation of the equation of mass conservation is employed. As a result, the diagonal components of the matrix form in the governing equations are not singular. The B-spline basis is used because it is superior to other splines in providing computer solutions to fluid flow problems. One of the advantages of the B-spline basis is that it has excellent approximation properties. Numerical examples of applications of the mixed formulation are presented to demonstrate the convergence characteristics and accuracy of the present formulation. © 1998 John Wiley & Sons, Ltd.     

不可压缩N-S方程的稳定分步算法及耦合离散

在有限增量微积分(finite increment calculus, FIC)的理论框架下，通过引入一个附加变量，发展了压力稳定型分步算法，有效改善了经典 分步算法的压力稳定性，同时还避免了标准FIC方法中存在的空间高阶导数的计算. 为保证 数值方法同时具有较快的计算速度和较好的健壮性，发展了有限元与无网格的耦合空间离散 方法. 该方案可在网格发生扭曲的区域采用无网格法空间离散以保证求解的精度和稳定性， 而在网格质量较好的区域以及本质边界上保留使用有限元法空间离散以提高计算效率和便于 施加本质边界条件. 方腔流考题的数值模拟结果突出地显示了所发展的压力稳定型分步算 法比经典分步算法具有更好的压力稳定性，能够有效消除速度-压力插值空间违反LBB条件而 导致的压力场的虚假数值振荡. 平面Poisseuille流动和一个典型型腔充填过程的数值模拟 结果, 表明了发展的耦合离散方案相对于单一的有限元法和单一的无网格法在综合考虑计 算效率和算法健壮性方面的突出优点.     

An extension of the SIMPLE based discontinuous Galerkin solver to unsteady incompressible flows

In this paper, we present a SIMPLE based algorithm in the context of the discontinuous Galerkin method for unsteady incompressible flows. Time discretization is done fully implicit using backward differentiation formulae (BDF) of varying order from 1 to 4. We show that the original equation for the pressure correction can be modified by using an equivalent operator stemming from the symmetric interior penalty (SIP) method leading to a reduced stencil size. To assess the accuracy as well as the stability and the performance of the scheme, three different test cases are carried out: the Taylor vortex flow, the Orr‐Sommerfeld stability problem for plane Poiseuille flow and the flow past a square cylinder. (1) Simulating the Taylor vortex flow, we verify the temporal accuracy for the different BDF schemes. Using the mixed‐order formulation, a spatial convergence study yields convergence rates of k + 1 and k in the L²‐norm for velocity and pressure, respectively. For the equal‐order formulation, we obtain approximately the same convergence rates, while the absolute error is smaller. (2) The stability of our method is examined by simulating the Orr–Sommerfeld stability problem. Using the mixed‐order formulation and adjusting the penalty parameter of the symmetric interior penalty method for the discretization of the viscous part, we can demonstrate the long‐term stability of the algorithm. Using pressure stabilization the equal‐order formulation is stable without changing the penalty parameter. (3) Finally, the results for the flow past a square cylinder show excellent agreement with numerical reference solutions as well as experiments. Copyright © 2015 John Wiley & Sons, Ltd.     

Accurate iteration-free mixed-stabilised formulation for laminar incompressible Navier–Stokes: Applications to fluid–structure interaction

Stabilised mixed velocity–pressure formulations are one of the widely-used finite element schemes for computing the numerical solutions of laminar incompressible Navier–Stokes. In these formulations, the Newton–Raphson scheme is employed to solve the nonlinearity in the convection term. One fundamental issue with this approach is the computational cost incurred in the Newton–Raphson iterations at every load/time step. In this paper, we present an iteration-free mixed finite element formulation for incompressible Navier–Stokes that preserves second-order temporal accuracy of the generalised-alpha and related schemes for both velocity and pressure fields. First, we demonstrate the second-order temporal accuracy using numerical convergence studies for an example with a manufactured solution. Later, we assess the accuracy and the computational benefits of the proposed scheme by studying the benchmark example of flow past a fixed circular cylinder. Towards showcasing the applicability of the proposed technique in a wider context, the inf–sup stable P2–P1 pair for the formulation without stabilisation is also considered. Finally, the resulting benefits of using the proposed scheme for fluid–structure interaction problems are illustrated using two benchmark examples in fluid-flexible structure interaction.     

Numerical instabilities and convergence control for convex approximation methods

Convex approximation methods could produce iterative oscillation of solutions for solving some problems in structural optimization. This paper firstly analyzes the reason for numerical instabilities of iterative oscillation of the popular convex approximation methods, such as CONLIN (Convex Linearization), MMA (Method of Moving Asymptotes), GCMMA (Global Convergence of MMA) and SQP (Sequential Quadratic Programming), from the perspective of chaotic dynamics of a discrete dynamical system. Then, the usual four methods to improve the convergence of optimization algorithms are reviewed, namely, the relaxation method, move limits, moving asymptotes and trust region management. Furthermore, the stability transformation method (STM) based on the chaos control principle is suggested, which is a general, simple and effective method for convergence control of iterative algorithms. Moreover, the relationships among the former four methods and STM are exposed. The connection between convergence control of iterative algorithms and chaotic dynamics is established. Finally, the STM is applied to the convergence control of convex approximation methods for optimizing several highly nonlinear examples. Numerical tests of convergence comparison and control of convex approximation methods illustrate that STM can stabilize the oscillating solutions for CONLIN and accelerate the slow convergence for MMA and SQP.     

A Nonlinear Galerkin/Petrov-Least Squares Mixed Element Method for the Stationary Navier-Stokes Equation

A nonlinear Galerkin/ Petrov- least squares mixed element (NGPLSME) method for the stationary Navier-Stokes equations is presented and analyzed. The scheme is that Petrov-least squares forms of residuals are added to the nonlinear Galerkin mixed element method so that it is stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. The existence, uniqueness and convergence ( at optimal rate ) of the NGPLSME solution is proved in the case of sufficient viscosity ( or small data).     

A nonlinear Galerkin/Petrov-least squares mixed element method for the stationary Navier-Stokes equations

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Polygonal finite elements for incompressible fluid flow

We discuss the use of polygonal finite elements for analysis of incompressible flow problems. It is well‐known that the stability of mixed finite element discretizations is governed by the so‐called inf‐sup condition, which, in this case, depends on the choice of the discrete velocity and pressure spaces. We present a low‐order choice of these spaces defined over convex polygonal partitions of the domain that satisfies the inf‐sup condition and, as such, does not admit spurious pressure modes or exhibit locking. Within each element, the pressure field is constant while the velocity is represented by the usual isoparametric transformation of a linearly‐complete basis. Thus, from a practical point of view, the implementation of the method is classical and does not require any special treatment. We present numerical results for both incompressible Stokes and stationary Navier–Stokes problems to verify the theoretical results regarding stability and convergence of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

SIMPLE‐type preconditioners for the Oseen problem

In this paper, the steady incompressible Navier–Stokes equations are discretized by the finite element method. The resulting systems of equations are solved by preconditioned Krylov subspace methods. Some new preconditioning strategies, both algebraic and problem dependent are discussed. We emphasize on the approximation of the Schur complement as used in semi implicit method for pressure‐linked equations‐type preconditioners. In the usual formulation, the Schur complement matrix and updates use scaling with the diagonal of the convection–diffusion matrix. We propose a variant of the SIMPLER preconditioner. Instead of using the diagonal of the convection–diffusion matrix, we scale the Schur complement and updates with the diagonal of the velocity mass matrix. This variant is called modified SIMPLER (MSIMPLER). With the new approximation, we observe a drastic improvement in convergence for large problems. MSIMPLER shows better convergence than the well‐known least‐squares commutator preconditioner which is also based on the diagonal of the velocity mass matrix. Copyright © 2008 John Wiley & Sons, Ltd.     

Approximation of generalized Stokes problems using dual‐mixed finite elements without enrichment

In this work a finite element method for a dual‐mixed approximation of generalized Stokes problems in two or three space dimensions is studied. A variational formulation of the generalized Stokes problems is accomplished through the introduction of the pseudostress and the trace‐free velocity gradient as unknowns, yielding a twofold saddle point problem. The method avoids the explicit computation of the pressure, which can be recovered through a simple post‐processing technique. Compared with an existing approach for the same problem, the method presented here reduces the global number of degrees of freedom by up to one‐third in two space dimensions. The method presented here also represents a connection between existing dual‐mixed and pseudostress methods for Stokes problems. Existence, uniqueness, and error results for the generalized Stokes problems are given, and numerical experiments that illustrate the theoretical results are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Discrete formulation of mixed finite element methods for vapor deposition chemical reaction equations

The vapor deposition chemical reaction processes, which are of extremely extensive applications, can be classified as a mathematical model by the following governing nonlinear partial differential equations containing velocity vector, temperature field, pressure field, and gas mass field. The mixed finite element (MFE) method is employed to study the system of equations for the vapor deposition chemical reaction processes. The semidiscrete and fully discrete MFE formulations are derived. And the existence and convergence (error estimate) of the semidiscrete and fully discrete MFE solutions are demonstrated. By employing MFE method to treat the system of equations for the vapor deposition chemical reaction processes, the numerical solutions of the velocity vector, the temperature field, the pressure field, and the gas mass field can be found out simultaneously. Thus, these researches are not only of important theoretical means, but also of extremely extensive applied vistas.     

Pressure‐stabilized maximum‐entropy methods for incompressible Stokes

We present a parameter‐free stable maximum‐entropy method for incompressible Stokes flow. Derived from a least‐biased optimization inspired by information theory, the meshfree maximum‐entropy method appears as an interesting alternative to classical approximation schemes like the finite element method. Especially compared with other meshfree methods, e.g. the moving least‐squares method, it allows for a straightforward imposition of boundary conditions. However, no Eulerian approach has yet been presented for real incompressible flow, encountering the convective and pressure instabilities. In this paper, we exclusively address the pressure instabilities caused by the mixed velocity‐pressure formulation of incompressible Stokes flow. In a preparatory discussion, existing stable and stabilized methods are investigated and evaluated. This is used to develop different approaches towards a stable maximum‐entropy formulation. We show results for two analytical tests, including a presentation of the convergence behavior. As a typical benchmark problem, results are also shown for the leaky lid‐driven cavity. The already presented information‐flux method for convection‐dominated problems in mind, we see this as the last step towards a maximum‐entropy method capable of simulating full incompressible flow problems. Copyright © 2015 John Wiley & Sons, Ltd.     

A spectral/hp least‐squares finite element analysis of the Carreau–Yasuda fluids

A least‐squares finite element model with spectral/hp approximations was developed for steady, two‐dimensional flows of non‐Newtonian fluids obeying the Carreau–Yasuda constitutive model. The finite element model consists of velocity, pressure, and stress fields as independent variables (hence, called a mixed model). Least‐squares models offer an alternative variational setting to the conventional weak‐form Galerkin models for the Navier–Stokes equations, and no compatibility conditions on the approximation spaces used for the velocity, pressure, and stress fields are necessary when the polynomial order (p) used is sufficiently high (say, p > 3, as determined numerically). Also, the use of the spectral/hp elements in conjunction with the least‐squares formulation with high p alleviates various forms of locking, which often appear in low‐order least‐squares finite element models for incompressible viscous fluids, and accurate results can be obtained with exponential convergence. To verify and validate, benchmark problems of Kovasznay flow, backward‐facing step flow, and lid‐driven square cavity flow are used. Then the effect of different parameters of the Carreau–Yasuda constitutive model on the flow characteristics is studied parametrically. Copyright © 2016 John Wiley & Sons, Ltd.     

Simulation of peristaltic flow of chyme in small intestine for couple stress fluid

In this article couple stress fluid have been considered for the peristaltic flow of chyme in intestine. Problem under consideration have been formulated assuming that two non-periodic sinusoidal wave of different wavelength propagate with same speed c along the outer wall of the tube. Governing equations have been simplified under the assumptions of long wavelength and low Reynolds number approximation (such assumption are consistent that Re (Reynold number) is very small and long wavelength approximation also exists in the small intestine). Exact solutions have been evaluated for velocity and pressure rise. Physical behaviour of different parameter of couple stress fluid have been presented graphically for velocity, pressure rise, pressure gradient and frictional forces. The stream lines are also made against different parameters.     

An ‘assumed deviatoric stress–pressure–velocity’ mixed finite element method for unsteady,convective, incompressible viscous flow: Part I: Theoretical development

A formulation of a mixed finite element method for the analysis of unsteady, convective, incompressible viscous flow is presented in which: (i) the deviatoric-stress, pressure, and velocity are discretized in each element, (ii) the deviatoric stress and pressure are subject to the constraint of the homogeneous momentum balance condition in each element, a priori, (iii) the convective acceleration is treated by the conventional Galerkin approach, (iv) the finite element system of equations involves only the constant term of the pressure field (which can otherwise be an arbitrary polynomial) in each element, in addition to the nodal velocities, and (v) all integrations are performed by the necessary order quadrature rules. A fundamental analysis of the stability of the numerical scheme is presented. The method is easily applicable to 3-dimensional problems. However, solutions to several problems of 2-dimensional Navier-Stokes' flow, and their comparisons with available solutions in terms of accuracy and efficiency, are discussed in detail in Part II of this paper.     

A face-based monolithic approach for the incompressible magnetohydrodynamics equations

A novel numerical algorithm has been developed to solve the incompressible resistive magnetohydrodynamics equations in a fully coupled form. The numerical method is based on the face-centered unstructured finite volume approximation, where the velocity and magnetic field vector components are defined at the center of edges/faces; meanwhile, the pressure term is defined at element centroid. In order to enforce a divergence-free magnetic field, the gradient of a scalar Lagrange multiplier is introduced into the induction equation. A special attention will be given to satisfy the continuity equation and the Gauss' law for magnetism within each element and the summation of the equations can be exactly reduced to the domain boundary. The first modification to the original algorithm involves the evaluation of the convective fluxes over the two neighboring elements, where the discrete continuity equations are exactly satisfied. The second modification is based on the neglecting electric field term from the Lorentz force in two dimensions. The resulting large-scale algebraic linear equations are solved in a fully coupled manner using the one- and two-level restricted additive Schwarz preconditioners to avoid any time step restrictions forced by stability requirements. The spatial convergence of the algorithm is confirmed by solving the Hartmann flow, and then the algorithm is applied to the classical lid-driven cavity and backward facing step benchmark problems in two and three dimensions. The lid-driven cavity flow calculations at relatively high Stuart numbers indicate the perfect braking effect of the magnetic field in two dimensions.     

Problems of Hydrodynamics for a Triaxial Ellipsoid

Problems of motion of a triaxial ellipsoid in an ideal liquid and in a viscous liquid in the Stokes approximation and also equilibrium shapes of the rotating gravitating liquid mass are considered. Solutions of these problems expressed via four quadratures depending on four parameters are significantly simplified because they are expressed via the only function of two arguments. The efficiency of the proposed approach is demonstrated by means of analyzing the velocity and pressure fields in an ideal liquid, calculating the added mass of the ellipsoid, determining the viscous friction, and studying the equilibrium shapes and stability of the rotating gravitating capillary liquid. The pressure on the triaxial ellipsoid surface is expressed via the projection of the normal to the impinging flow velocity. The shape of an ellipsoid that ensures the minimum viscous drag at a constant volume is determined analytically. A simple equation in elementary functions is derived for determining the boundary of the domains of the secular stability of the Maclaurin ellipsoids. An approximate solution of the problem of equilibrium and stability of a rotating droplet is presented in elementary functions. A bifurcation point with non-axisymmetric equilibrium shapes branching from this point is found.     

The cause and cure (!) of the spurious pressures generated by certain fem solutions of the incompressible Navier-Stokes equations: Part 2

The spurious pressures and ostensibly acceptable velocities which sometimes result from certain FEM approximate solutions of the incompressible Navier-Stokes equations are explained in detail. The concept of pressure modes, physical and spurious, pure and impure, is introduced and their effects on discretized solutions is analysed, in the context of mixed interpolation and penalty approaches. Pressure filtering schemes, which are capable of recovering useful pressures from otherwise polluted numerical results, are developed for two particular elements in two-dimensions and one element in three-dimensions. The automatic pressure filter associated with the penalty method is also explained. Implications regarding the effect of spurious pressure modes on accuracy and ultimate convergence with mesh refinement are discussed and a list of unanswered questions presented. Sufficient numerical examples are discussed to corroborate the theory presented herein.