The least-squares finite element method for low-mach-number compressible viscous flows

The present paper reports the development of the Least-Squares Finite Element Method (LSFEM) for simulating compressible viscous flows at low Mach numbers in which the incompressible flows pose as an extreme. The conventional approach requires special treatments for low-speed flows calculations: finite difference and finite volume methods are based on the use of the staggered grid or the preconditioning technique, and finite element methods rely on the mixed method and the operator-splitting method. In this paper, however, we show that such a difficulty does not exist for the LSFEM and no special treatment is needed. The LSFEM always leads to a symmetric, positive-definite matrix through which the compressible flow equations can be effectively solved. Two numerical examples are included to demonstrate the method: driven cavity flows at various Reynolds numbers and buoyancy-driven flows with significant density variation. Both examples are calculated by using full compressible flow equations.     

Thermal analysis of the torsion test under hot-working conditions

It is important to estimate the temperature changes in material testing so that the interpretation of test data can be improved and the derived constitutive equations validated. In this paper a finite element thermal analysis of the torsion test under hot-working conditions is carried out and the flow stresses of an aluminum alloy AA5252 are predicted and compared with the data. It is shown that the further flow softening after the peak stresses have been obtained, which is attributed to excessive heat accumulation during testing at high strain rates, can be predicted reasonably by the method presented.     

The extended finite element method for rigid particles in Stokes flow

A new method for the simulation of particulate flows, based on the extended finite element method (X‐FEM), is described. In this method, the particle surfaces need not conform to the finite element boundaries, so that moving particles can be simulated without remeshing. The near field form of the fluid flow about each particle is built into the finite element basis using a partition of unity enrichment, allowing the simple enforcement of boundary conditions and improved accuracy over other methods on a coarse mesh. We present a weak form of the equations of motion useful for the simulation of freely moving particles, and solve example problems for particles with prescribed and unknown velocities. Copyright © 2001 John Wiley & Sons, Ltd.     

基于协调翘曲场的开闭口混合薄壁截面杆件约束扭转一维有限元分析

开口及闭口薄壁杆件约束扭转问题已由经典Timoshenko和Benscoter理论解决。然而,开闭口混合薄壁截面杆件约束扭转分析必须考虑开、闭口部分翘曲能力的差异,翘曲剪流形成机理有待进一步研究。该文假定开、闭口截面翘曲分别满足Vlasov和Umanskii假定,考虑开、闭口截面公共节点翘曲连续性要求,建立含有待定翘曲参数的协调翘曲模型。由截面受力平衡,确定翘曲参数显式列式,提出开闭口混合薄壁截面杆件约束扭转分析的一维有限元模型。算例及参数分析结果表明,基于Umanskii第二理论的I类方法在悬臂板及闭口周边引入附加剪流,影响翘曲剪应力精度。基于Umanskii第二理论的II类方法只能计算截面板件平均剪应力,无法反映真实翘曲剪流分布。基于Vlasov约束扭转假定的Beam-189单元忽略闭口周边约束效应产生的附加翘曲及剪流,影响翘曲正应力和剪应力精度。该文方法与Shell-63单元能得到基本吻合的变形与应力结果,说明一维梁元能正确反映开闭口混合薄壁截面杆件约束扭转及翘曲刚度。     

The numerical solution of two-dimensional,steady flow problems by the finite element method

Finite element equivalents of the equations governing shearing and buoyancy driven flows are derived, and reduced to upwind forms suitable for the solution of problems in which the Reynolds and Rayleigh numbers are large. A modification to the central difference iterative method is studied which increases the Reynolds and Rayleigh numbers for which a central difference form may be used. A comparison is made between the results obtained using the central and upwind forms of the finite element method and those predicted by finite difference methods in the case of flow in a cavity. A mesh refinement study is made. The upwind forms of the finite element equations are applied to the solution of a complex flow problem involving the flow of glass in a throated furnace in the case of constant- and temperature- dependent viscosity and conductivity.     

The partition of unity finite element method for short wave acoustic propagation on non‐uniform potential flows

A novel numerical method is proposed for modelling time‐harmonic acoustic propagation of short wavelength disturbances on non‐uniform potential flows. The method is based on the partition of unity finite element method in which a local basis of discrete plane waves is used to enrich the conventional finite element approximation space. The basis functions are local solutions of the governing equations. They are able to represent accurately the highly oscillatory behaviour of the solution within each element while taking into account the convective effect of the flow and the spatial variation in local sound speed when the flow is non‐uniform. Many wavelengths can be included within a single element leading to ultra‐sparse meshes. Results presented in this article will demonstrate that accurate solutions can be obtained in this way for a greatly reduced number of degrees of freedom when compared to conventional element or grid‐based schemes. Numerical results for lined uniform two‐dimensional ducts and for non‐uniform axisymmetric ducts are presented to indicate the accuracy and performance which can be achieved. Numerical studies indicate that the ‘pollution’ effect associated with cumulative dispersion error in conventional finite element schemes is largely eliminated. Copyright © 2005 John Wiley & Sons, Ltd.     

A method for incorporating free boundaries with surface tension in finite element fluid-flow simulators

A boundary-location method is developed for finite element simulation of steady, two-dimensional flows of Newtonian liquid with free boundaries. In the method, boundary shape and position and the velocity and pressure fields are determined simultaneously. Inertial, viscous, gravitational, and surface tension effects are included in the development. The complete set of nonlinear finite element equations is solved by a modified frontal method combined with Newton-Raphson iteration to speed convergence. The finite element used to illustrate the method approximates the pressure as a piecewise constant function and the velocity and free boundaries as piecewise linear functions. Example calculations for flow from a slit show that the method can be effective.     

A Treatment of wall boundaries for turbulent flows by the use of a transmission finite element method

A method to connect momentum Navier-Stokes equations with the universal law of the wall using the finite element method is developed for turbulent wall flows. This method is based on a domain decomposition of the fluid into subdomains near a solid boundary where the law of the wall is valid. A transmission formulation is introduced to match these regions and a new class of boundary finite element is used. This finite element takes into account the near-wall profile of the velocity and the transmission conditions. Computational results are presented for Poiseuille flow and flow over a backward-facing step.     

A block iterative finite element algorithm for numerical solution of the steady–state,compressible Navier–Stokes equations

An iterative method for numerically solving the time independent Navier–Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss–Seidel principle in block form to the systems of the non-linear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C⁰-cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and symptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.     

Hybrid finite element/volume method for shallow water equations

A hybrid numerical scheme based on finite element and finite volume methods is developed to solve shallow water equations. In the recent past, we introduced a series of hybrid methods to solve incompressible low and high Reynolds number flows for single and two‐fluid flow problems. The present work extends the application of hybrid method to shallow water equations. In our hybrid shallow water flow solver, we write the governing equations in non‐conservation form and solve the non‐linear wave equation using finite element method with linear interpolation functions in space. On the other hand, the momentum equation is solved with highly accurate cell‐center finite volume method. Our hybrid numerical scheme is truly a segregated method with primitive variables stored and solved at both node and element centers. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing which will act only around sharp interfaces without sacrificing the accuracy elsewhere. Matrix‐free GMRES iterative solvers are used to solve both the wave and momentum equations in finite element and finite volume schemes. Several test problems are presented to demonstrate the robustness and applicability of the numerical method. Copyright © 2010 John Wiley & Sons, Ltd.     

Computations of high‐speed compressible flows with adaptive cell‐centered finite element method

Abstract

An upwind cell‐centered finite element formulation is combined with an adaptive meshing technique to solve Navier‐Stokes equations for high‐speed inviscid and viscous compressible flows. The finite element formulation and the computational procedure are described. An adaptive meshing technique is applied to increase the analysis solution accuracy, as well as to minimize the computational time and the computer memory requirement. The efficiency of the combined method is evaluated by the examples of Mach 2.6 inviscid flow in a channel with compression and expansion ramps, Mach 6.47 inviscid and viscous flows past a cylinder, and Mach 4 viscous flow over a flat plate.     

Derivation of a new stiffness matrix for helically armoured cables considering tension and torsion

A new element stiffness matrix is derived for straight cable elements subjected to tension and torsion. The cross-section of a cable, which may consist of many different structural components, is treated in the following as a single composite element. The derivation is quite general; consequently, the results can be used for a broad category of cable configurations. Individual helical armourning wires, for instance, may have unique geometric and material properties. In addition, no limit is placed on the number of wire layers. Furthermore, compressibility of the central core element can also be considered. The equations of equilibrium are first derived to include ‘internal’ geometric non-linearties produced by large deformations (axial elongation and rotatioin) of a straight cable element. These equations are then linearized in a consistent manner to give a liner stiffness matrix. Linear elasticity is assumed throughout. Excellent agreement with experimental results for two different cables validates the correctness of the analysis.     

基于有限元分析的直井中钻柱螺旋屈曲临界载荷定义

采用有限元法对直井中钻柱非线性屈曲控制微分方程进行了求解,力学模型中考虑了钻柱的重力,摒弃了传统分析中的无重力、等螺距和小位移假设,考察了不同边界条件对钻柱屈曲的影响。基于有限元分析的结果给出了钻柱非线性螺旋屈曲临界载荷定义,分析了位移高阶项在钻柱弯矩计算中的影响。分析表明,根据给出的定义确定的钻柱螺旋屈曲临界载荷与实验数据吻合,位移高阶项在弯矩计算中不可忽略。为石油钻采工程中钻柱螺旋屈曲临界载荷的预测提供了一种有效的方法。     

粘弹流动的间断有限元模拟

针对传统有限元法求解Oldroyd-B本构方程时需加入稳定化方案的缺点,本文基于非结构网格给出了统一间断有限元求解框架.该框架包含采用IPDG(interior penalty discontinuous Galerkin)求解质量方程和动量方程,与采用RKDG(RungeKutta discontinuous Galerkin)求解本构方程这两个核心.数值结果表明:该方法在求解Oldroyd-B本构方程时无需加入稳定化方案,实施比有限元法简便,且具有较高的计算精度,可有效地模拟包含应力奇异点的复杂粘弹流动问题,进而揭示非牛顿粘弹流动的基本特征.     

An efficient semi-implicit finite element scheme for two-dimensional tidal flow computations

A semi-implicit finite element scheme is proposed for two-dimensional tidal flow computations. In the scheme, each term of the governing equations, rather than each dependent variable, is expanded in terms of the unknown nodal values and it helps to reduce computer execution time. The friction terms are represented semi-implicitly to improve stability, but this requires no additional computational effort. Test cases where analytic solutions have been obtained for the shallow water equations are employed to test the proposed scheme and the test results show that the scheme is efficient and stable. An numerical experiment is also included to compare the proposed scheme with another finite element scheme employing Serendipity-type Hermitian cubic basis functions. A numerical model of an actual bay is constructed based on the proposed scheme and computed tidal flows bear close resemblance to flows measured in field survey.     

A coupled two degree of freedom pull-in model for micromirrors under capillary force

The current paper presents a two degree of freedom model for the problem of micromirrors under capillary force. The principal of minimum potential energy is employed for finding the equilibrium equations governing the deflection and the rotation of the micromirror. Then, using the implicit function theorem, a coupled bending–torsion model is presented for pull-in characteristics of micromirrors under capillary force and the concept of instability mode is introduced. It is observed that with increasing ratio of bending and torsion stiffness, the dominant instability mode changes from bending mode to the torsion mode. In order to verify the accuracy of the coupled model, static behavior of a group of micromirrors is investigated both analytically using the presented model and numerically using the commercial finite element software ANSYS. It is observed that results of the coupled model match well with the results of finite element simulations, but they both deviate considerably from the results of the pure torsion model. The presented coupled model can be used for safe and stable design of micromirrors under capillary force.     

Fluid Flow and Solidification Under Combined Action of Magnetic Fields and Microgravity

Mathematical models, both 2-D and 3-D, are developed to represent g-jitter induced fluid flows and their effects on solidification under combined action of magnetic fields and microgravity. The numerical model development is based on the finite element solution of governing equations describing the transient g-jitter driven fluid flows, heat transfer, and solutal transport during crystal growth with and without an applied magnetic field in space vehicles. To validate the model predictions, a ground-based g-jitter simulator is developed using the oscillating wall temperatures where timely oscillating fluid flows are measured using a laser PIV (Particle Image Velocimetry) system. The measurements are compared well with numerical results obtained from the numerical models. Results show that a combined action derived from magnetic damping and microgravity can be an effective means to control the melt flow and solutal transport of single crystal growth in space environment.     

Theory and numerics of three‐dimensional beams with elastoplastic material behaviour

A theory of space curved beams with arbitrary cross‐sections and an associated finite element formulation is presented. Within the present beam theory the reference point, the centroid, the centre of shear and the loading point are arbitrary points of the cross‐section. The beam strains are based on a kinematic assumption where torsion‐warping deformation is included. Each node of the derived finite element possesses seven degrees of freedom. The update of the rotational parameters at the finite element nodes is achieved in an additive way. Applying the isoparametric concept the kinematic quantities are approximated using Lagrangian interpolation functions. Since the reference curve lies arbitrarily with respect to the centroid the developed element can be used to discretize eccentric stiffener of shells. Due to the implemented constitutive equations for elastoplastic material behaviour the element can be used to evaluate the load‐carrying capacity of beam structures. Copyright © 2000 John Wiley & Sons, Ltd.     

A ψ–ω finite element formulations for the Navier-Stokes equations

A new method of solving the two-dimensional Navier–Stokes equations by using the finite element method is proposed. The flow is represented by the stream function–vorticity formulation and the no-slip boundary conditions are explicitly introduced in the nonlinear equations. This formulation coupled with the Newton-Raphson method enables the study of stationary flows for high Reynolds number, without any convergence problem. A number of flow problems are analysed in order to demonstrate the efficiency of the present formulation.     

A new least-squares finite element model for combined Navier–Stokes and Darcy flows in geometrically complicated domains with solid and porous boundaries

In this paper we present a novel method for linking Navier–Stokes and Darcy equations along a porous inner boundary in a flow regime which is governed by both types of these equations. The method is based on a least-squares finite element technique and uses isoparametric C¹ continuous Hermite elements for domain discretization. We show that our technique is superior to previously developed models for the combined Navier–Stokes/Darcy flows. The previous works use weighted residual finite element procedures in conjunction with C⁰ elements which are inherently incapable of linking Navier–Stokes and Darcy equations. The paper includes the application of our model to a geometrically complicated axisymmetric slurry filtration system.