A finite difference-Galerkin finite element solution of compressible laminar boundary-layer flows

A finite difference-Galerkinfinite element method is presented for the solution of the two-dimensional compressible laminar boundary-layer flow problem. The streamwise derivatives in the momentum and energy equations are approximated by finite differences. An iterative scheme, due to the non-linearity of the problem, in conjunction with the Galerkin finite element method is then proposed for the solution of the problem through the boundary-layer thickness. Numerical results are presented and these are compared with other numerical and analytical solutions in order to show the applicability and the effectiveness of the proposed formulation. In all the cases here examined, the results obtained attained the same accuracy of other numerical methods for a much smaller number of points in the boundary-layer.     

COMPARISON OF ANALYTICAL, NUMERICAL AND EXPERIMENTAL SOLUTIONS TO PROBLEMS OF DEEPLY CRACKED WELDED JOINTS IN BENDING

This paper presents analytical and numerical solutions to the analysis of welded specimens when loaded in three-point bending, and compares the results with those obtained experimentally. In each case the crack is located within the weld material, and runs parallel to the weld. Two analytical models are presented for deriving limit loads using slip-line field theory. Due to the welding process, the material behaviour in the weld-base interface is complex, and this is described in the analytical solutions using a number of material zones. The analytical solutions also provide the η and d_∞ coefficients which are used to determine the J and crack tip opening displacement (CTOD) parameters. These solutions are then compared with numerical results obtained using the finite element method. Good agreement is obtained between the numerical and analytical results, and it is shown that for overmatched specimens a better analytical solution can be obtained by using a slip-line field geometry which passes predominantly through the base material. When the analytical η solutions are applied to the experimental results it is shown that, in the case of undermatched weldments, J can be used as a fracture characterising parameter, but it cannot be used in the case of an overmatched specimen.     

One-dimensional finite element method in hydrodynamic stability

The stability of plane Poiseuille flow and circular Couette flow are examined with respect to linear azimuthally periodic disturbances by the finite element method. In the case of Couette motion, solutions are obtained for a narrow gap, a wide gap and a dilute polymer solution with an elongational viscosity in the narrow gap limit when both cylinders rotate at almost equal speed in the same direction. Results are in good agreement with previous calculations by other numerical methods.     

The finite element method applied to ideal gaseous mixtures

The finite element technique is extended to an ideal gaseous mixture to study diffusion of several gaseous species in a carrier gas under various geometries and flow conditions. This is accomplished by obtaining the finite element analog of the balance of mass for each species, along with the corresponding equations for the balance of mass, momentum and energy for the mixture. Numerical results are obtained for a steady-state, incompressible, isothermal, viscous mixture of two gases for several boundary conditions. Favourable agreement is obtained between the finite element results and analytical solutions.     

Finite deformation finite element analyses of tensile growing crack fields in elastic-plastic material

Finite deformation finite element analyses of plane strain stationary and quasi-statically growing crack fields in fully incompressible elastic-ideally plastic material are reported for small-scale yielding conditions. A principal goal is to determine the differences between solutions of rigorous finite deformation formulation and those of the usual small-displacement-gradient formulation, and thereby assess the validity of the (nearly all) extant studies of ductile crack growth that are based on a small-displacement-gradient formulation. The stationary crack case with a significantly blunted tip is studied first; excellent agreement in stress characteristics at all angles about the crack tip and up to a radius of about three times the crack tip opening displacement is shown between Rice and Johnson's [1] approximate analytical solution and our numerical solution. Outside this radius, the numerical results agree very well with Drugan and Chen's [2] small-displacement-gradient analytical characteristics solution in the region of principal plastic deformation. Thus we identify accurate analytical representations for the stress field throughout the plastic zone of a blunted stationary crack. For the growing crack case, the macroscopic difference in crack tip opening profiles between previous small-displacement-gradient solutions and the present results is shown to be negligible, as is the difference in the stress fields in plastic regions. The stress characteristics again agree very well with analytical results of [2]. The numerical results suggest—in agreement with a recent analytical finite deformation study by Reid and Drugan [3]—that it is the finite geometry changes rather than the additional spin terms in the objective constitutive equation that cause any differences between the small-displacement-gradient and the finite deformation solutions, and that such differences are nearly indistinguishable for growing cracks.     

冲击荷载下K8单层球面网壳的破坏类型

首先介绍求解冲击问题解析法与数值法的算法基础,以及有限元求解的基本解法(中心差分法)。之后介绍了以上述为理论基础的计算冲击荷载的国际通用有限元软件ANSYS/LS-DYNA。并分别用解析法与有限元软件求解同一简单模型,互相校核;二者结论接近,验证了软件求解功能及相关参数设置的正确性。随后在ANSYS/LS-DYNA软件中建立了60m跨度K8型单层球面网壳抗冲击模型,在上述基础上,通过对135个算例的分析总结,根据竖向变形与塑性发展情况总结了K8单层球面网壳结构在顶点冲击荷载下的五种破坏类型及其分布规律。     

Thermodynamic considerations in the numerical simulation of steady compressible flow

The notion of thermodynamic consistency of a finite difference scheme is introduced in connection with steady compressible flow simulation. This concept requires that the finite difference scheme must lead to solutions which are consistent with the Second Law of Thermodynamics, whatever the grid sizes employed. Thus, in addition to the usual consistency with the conservation equations of mass, momentum and energy, the solutions of the finite difference equations must satisfy thermodynamic restrictions. The scheme proposed by Singhal and Spalding for the computation of plane, isentropic flow is generalized in a thermodynamically consistent way to handle inviscid, adiabatic but non-isentropic plane flow. Numerical solutions are obtained for a model problem using both the original and the extended methods. The results are compared with analytical predictions of shock-expansion theory. The effectiveness and the thermodynamic consistency of the new formulation is demonstrated.     

DIRECT FORMULATION FINITE ELEMENT METHOD FOR TWO-DIMENSIONAL GROUNDWATER POLLUTION MODELLING WITH A COMPARISON WITH CONVENTIONAL FINITE ELEMENT METHOD

Real world ground water pollution modelling deals with solute transport through anisotropic, heterogeneous media. The applicability of analytical solutions for such a real world system is extremely limited. As an effective tool, numerical models, such as finite difference and finite element methods, are usually employed to model field scenarios. Nevertheless, ground water pollution modelling is a hallenging task and frequently ends up with misleading results. Most of the time insufficient data are blamed for such erratic results. A recent investigation shows that the shortcomings of numerical formulations may be the major cause for many disputes and confusions in numerical analyses. In reality, a point injection of water in a static, homogeneous and isotropic groundwater system shows a radial dissipation of water forming a sphere; and a full-depth line injection shows a radial dissipation forming a cylinder. The finite difference method completely ignores this fundamental flow principles and allows water only to flow along orthogonal directions. To overcome this limitation, the finite element method was developed as a flexible approach in order to connect a node with the neighbouring nodes in various directions where water is assumed to flow in any directions along node connections. In a recent investigation, it has been found that the conventional finite element method does not keep the commitments; and its formulation techniques lead to a global matrix where a solution domain is not connected with all the neighbouring nodes and does not comply with the control-volume mass balance concept. A consistent finite element formulation approach which does not need imaginary mathematical formulation and overcomes the limitations of both the conventional finite difference and finite element methods has been developed. This method allows fluid flow and solute transport in a porous medium in radial directions. The global matrices for flow and transport obtained from this technique are field representative, diagonally dominant and easily convergent. The new method is robust, needs less mathematical computation and has many advantages over the conventional finite difference and finite element methods.     

基于剪切耗能假设的黏弹性夹芯梁的振动和阻尼特性

基于一阶剪切变形理论和哈密顿原理建立了三层粘弹性夹芯梁结构的有限元模型并对其振动和阻尼特性进行了研究。建模时认为粘弹材料层不可压缩,振动能量是依靠粘弹性层的剪切变形来耗散的。为验证本模型的正确性,将其与解析解作了对比。同时,为了证明本方法的优越性,将其与常用的“实特征模态”、“近似复特征模态”、“钻石法”和“近似法”四种数值方法做了比较。结果表明本方法的精度在这几种数值方法中是最好的。最后,讨论了粘弹性夹芯梁结构参数变化对系统固有频率和损耗因子的影响,得到了一些有工程实际意义的结论。     

Finite Element Analysis of Nonlinear Fully Coupled Thermoelectric Materials

A complete nonlinear coupled finite element algorithm for thermoelectric materials is developed and implemented within a two-dimensional finite element code. Starting from a suitable formulation of the constitutive (including Seebeck, Joule, Peltier and Thompson effects) and equilibrium equations, residual vectors and consistent tangent matrices are formulated and implemented in a standard four-node isoparametric element with two degrees of freedom per node (voltage and temperature). The nonlinearities arise due to the coupling between the electric and thermal fluxes, to the dependence of the discretized material parameters with the independent variable temperature and to a convective-type term, representing the thermal energy that electrons carry. Three examples for Bi₂Te₃ thermoelements are presented. The first two compare the numerical results with simplified, one-dimensional analytical solutions. The first example is related to a linear uncoupled evaluation, the second to a nonlinear coupled Seebeck effect. Perfect agreement is obtained between the analytical (obtained by considering the material properties constant in the second case) and numerical solutions. The first two examples show that the dependency of the material coefficents is not important for one-dimensional cases. The third example, without direct analytical solution due to the complete coupling and material nonlinearity, studies the two-dimensional Peltier effect of a thermopair.     

Higher-order and higher floating-point precision numerical approximations of finite strain elasticity moduli

Two real-domain numerical approximation methods for accurate computation of finite strain elasticity moduli are developed and their accuracy and computational efficiency are investigated, with reference to hyperelastic constitutive models with known analytical solutions. The methods are higher-order and higher floating-point precision numerical approximation, the latter being novel in this context. A general formula for higher-order approximation finite difference schemes is derived and a new procedure is proposed to implement increased floating-point precision. The accuracy of the approximated elasticity moduli is investigated numerically using higher-order approximations in standard double precision and increased quadruple precision. It is found that, as the order of the approximation increases, the elasticity moduli tend toward the analytical solution. Using higher floating-point precision, the approximated elasticity moduli for all orders of approximation are found to be more accurate than the standard double precision evaluation of the analytical moduli. Application of the techniques to a finite element problem shows that the numerically approximated methods obtain convergence equivalent to the analytical method but require greater computational effort. It is concluded that numerical approximation of elasticity moduli is a powerful and effective means of implementing advanced constitutive models in the finite element method without prior derivation of difficult analytical solutions.     

Determination of stress intensity factors in cracked plates by the finite element method

Stress fields near crack tips in an elastic body can be specified by the stress intensity factors which are closely related to the stress singularities arising from the crack tips. These singularities, however, cannot be represented exactly by conventional finite element models. A new method for the analysis of stresses around cracks is proposed in this paper on the basis of the superposition of analytical and finite element solutions. This method is applied to several two-dimensional problems whose solutions are obtained analytically, and it is shown that their numerical results are in excellent agreement with analytical ones. Sufficiently accurate results can be obtained by the conventional finite element analysis with rather coarse mesh subdivision. Computational efforts are then considerably reduced compared with other methods.     

An alternative analytical solution for water-wave motion over a submerged horizontal porous plate

This study gives an alternative analytical solution for water-wave motion over an offshore submerged horizontal porous-plate breakwater in the context of linear potential theory. The matched-eigenfunction-expansions method is used to obtain the solution. The solution consists of a symmetric part and an antisymmetric part. The symmetric part is also the solution of wave reflection by a vertical solid wall with a submerged horizontal porous plate attached to it. In comparison with previous analytical solutions with respect to finite submerged horizontal porous plates, no complex water-wave dispersion relations are included in the present solution. Thus, the present solution is easier for numerical implementation. Numerical examples show that the convergence of the present solution is satisfactory. The results of the present solution also agree well with previous results by different analytical approaches, as well as previous numerical results by different boundary-element methods. The present solution can be easily extended to the case of multi-layer submerged horizontal porous plates, which may be more significant in practice for meeting different tide levels.     

Momentum Conservative Schemes for Shallow Water Flows

We discuss the implementation of the finite volume method on a staggered grid to solve the full shallow water equations with a conservative approximation for the advection term. Stelling & Duinmeijer [15] noted that the advection approximation may be energy-head or momentum conservative, and if suitable which of these to implement depends upon the particular flow being considered. The momentum conservative scheme pursued here is shown to be suitable for 1D problems such as transcritical flow with a shock and dam break over a rectangular bed, and we also found that our simulation of dam break over a dry sloping bed is in good agreement with the exact solution. Further, the results obtained using the generalised momentum conservative approximation for 2D shallow water equations to simulate wave run up on a conical island are in good agreement with benchmark experimental data.     

Mass conservation in numerical simulation of resin flow

The finite element/control volume (FE/CV) approach is commonly used for numerical simulation of resin flow in many composites manufacturing processes. The law of mass conservation is sometimes violated with the use of this approach. Especially, when the Galerkin formulation is used with isoparametric finite elements to obtain the pressure field, the balance of resin mass cannot be achieved.In this paper, reasons leading to such mass imbalance are investigated. A numerical model based on material incompressibility is developed to eliminate the problem. A few isothermal flow simulation examples are presented to demonstrate the application of the model. The results obtained are in excellent agreement with the analytical solutions.     

Finite volume analysis of laminated composite plates

A numerical procedure for analysis of general laminated plates under transverse load is developed utilizing the Mindlin plate theory, the finite volume discretization, and a segregated solution algorithm. The force and moment balance equations with the laminate constitutive relations are written in the form of a generic transport equation. In order to obtain discrete counterparts of the governing equations, the plate is subdivided into N control volumes by a Cartesian numerical mesh. As a result, five sets of N linear equations with N unknowns are obtained and solved using the conjugate gradient method with preconditioning. For the method validation, a number of test cases are designed to cover thick and thin laminated plates with aspect ratio (width to thickness) from 4 to 100. Simply supported orthotropic, symmetric cross‐ply, and angle‐ply laminated plates under uniform and sinusoidal pressure loads are solved, and results are compared with available analytical solutions. The shear correction factor of 5/6 is utilized throughout the procedure, which is consistent with test cases used in the reviewed literature. Comparisons of the finite volume method results for maximum deflections at the center of the plate and the Navier solutions obtained for aspect ratios 10, 20, and 100 shows a very good agreement. Copyright © 2016 John Wiley & Sons, Ltd.     

Comparison of method of lines and finite difference solutions of 2‐D Navier–Stokes equations for transient laminar pipe flow

Performances of method of lines (MOL) and finite difference method (FDM) were tested from the viewpoints of solution accuracy and central processing unit (CPU) time by applying them to the solution of time‐dependent 2‐D Navier–Stokes equations for transient laminar flow without/with sudden expansion and comparing their results with steady‐state numerical predictions and measurements previously reported in the literature. Predictions of both methods were obtained on the same computer by using the same order of spatial discretization and the same uniform grid distribution. Axial velocity and pressure distribution in pipe flow and steady‐state reattachment lengths in sudden expansion flow on uniform grid distribution predicted by both methods were found to be in excellent agreement. Transient solutions of both methods for pipe flow problem show favourable comparison and are in accordance with expected trends. However, non‐physical oscillations were produced by both methods in the transient solution of sudden expansion pipe flow. MOL was demonstrated to yield non‐oscillatory solutions for recirculating flows when intelligent higher‐order discretization scheme is utilized for convective terms. MOL was found to be superior to FDM with respect to CPU and set‐up times and its flexibility for incorporation of other conservation equations. Copyright © 2001 John Wiley & Sons, Ltd.     

一类平带驱动系统非线性振动的幅频特性

采用解析和数值方法得到了一类平带驱动系统非线性振动的幅频特性.考虑由主动轮、从动轮、张紧轮和张紧臂构成的平带驱动系统基本力学模型,建立了系统的振动方程.推广平均法的基本思想到多自由度系统,导出了平均化方程,进而可得到了系统响应的幅频特性.针对具体算例,通过用数值方法直接积分系统振动方程,得到了系统的幅频特性.解析结果与数值结果定性一致,但也存在定量方面的误差.     

功能梯度曲梁弯曲问题的解析解

该文采用弹性力学逆解法,求得了功能梯度曲梁在端部受弯矩作用的解析解。假设弹性模量E=E₀rⁿ沿径向呈幂函数的梯度分布。根据弹性力学平面问题的基本方程,在极坐标系下,引入应力函数,得到了弯曲问题的解析解。进而将功能梯度曲梁问题进行扩展,求得了整环或厚壁圆筒以及向错问题的解析解。将所得到的解退化到均匀弹性情况,与经典的理论解一致。最后对梯度函数按幂函数变化的算例进行了分析,结果显示梯度因子n对应力及位移的分布产生了巨大的影响。该文所得到的结论可以作为功能梯度曲梁构件优化设计的理论基础。     

A two-layer model for a non-Newtonian gravity current subjected to wind shear

Summary.  A theoretical model is developed for the gravity current resulting from the sudden release of a fixed volume of fluid of non-Newtonian power law rheology on top of a slightly denser Newtonian fluid layer in the presence of wind stress. The model incorporates the flow of both layers and accounts for the effects of inertial and viscous forces, and is suited for moderate Reynolds number flows. The governing equations are obtained by depth-averaging the unsteady equations of motion in accordance with the von Kármán's momentum integral method, and constitute a hyperbolic system of four equations for the flow rates and thicknesses of the fluid layers. Results are obtained by a well established numerical scheme for systems of nonlinear hyperbolic equations. For a particular case analytical results are obtained by employing an asymptotic matching approach. Good agreement is obtained between the numerical and analytical results. The effects of the thickness of the ambient layer, wind stress, Reynolds numbers, and rheology on the gravity current are discussed. Received July 22, 2002; revised November 27, 2002 Published online: May 8, 2003