复合材料层合圆柱壳的有限元分析

本文根据Reissner-Mindlin型的全局位移场(一阶和三阶)，应用有限元预测一修正法，数值计算和分析了机械载荷作用下复合材料层合圆柱壳的挠度和横向剪应力。首先按照一般的有限元分析过程(没有引入剪切修正系数)计算出层合圆柱壳的挠度预测值；然后利用Lagrange插值构造横向剪应力的一般形式，使得满足层间连续和表面上为零的条件，通过最小二乘法拟合三维应力平衡方程获得横向剪应力；最后在单元上计算和引入剪切修正系数，再经过有限元分析计算出层合圆柱壳的挠度修正值。数值计算结果与三维线弹性解的比较表明，挠度修正值和横向剪应力的精度是十分满意的。     

流体力学两种格式能量守恒比较

Lagrange系统下的非定常流体力学数值方法中，使用非守恒型能量方程获得的总能量(内能与动能之和)的误差大小是鉴别一种格式好坏的重要标志之一．讨论在校坐标系下两种有限元方法的离散格式及其能量守恒性．一种是采用由因子γ^-1来加权插值基函数的Galerkin有限元方法，即面平均格式；另一种是直接加权插值基函数的Galerkin有限元方法，即体平均格式．误差分析表明体平均格式具有较小的能量守恒误差，数值计算结果也显示出体平均格式能量守恒误差比面平均格式明显小．     

紧支试函数加权残值法

将紧支函数引入加权残值法中，提出了紧支试函数加权残值法，其数值格式具有和有限元相似的窄带系数矩阵，提高了加权残值法的计算效率．在紧支试函数加权残值的基础上，导出了紧支试函数直接配点法、紧支试函数Hermite配点法和紧支试函数最小二乘配点法的具体格式，并且对几个典型算例进行了分析．与配点法相比，这些方法精度高，稳定性好，而与Galerkin法相比，这些方法效率高．     

微重环境中三维液体非线性晃动的数值模拟

将ALE（任意的拉格朗日-欧拉）运动学描述关系引入到Navier-Stokes方程中,在时间域上采用分步离散方法中的速度修正格式,利用Galerkin加权余量方法推导了系统的有限元数值离散方程;推导了考虑表面张力效应时有限元边界件的弱积分形式。模拟了考虑表面张力情况下圆筒形贮腔中液体的非线性晃动,揭示了考虑表面张力效应时液体非线性晃动的重要特征。     

加权最小二乘无网格法

在最小二乘法和移动最小二乘近似的基础上提出了加权最小二乘无网格法．该方法除节点外又引入了一些辅助点，控制方程在所有节点和辅助点处的残差用最小二乘法予以消除，边界条件用罚函数法引入．另外对移动最小二乘近似进行了改进，并给出了最小二乘法中泛函的简化格式，因而提高了计算效率．与配点法相比，新方法精度高，稳定性好，并且系数矩阵是对称正定矩阵．与Galerkin法相比，该方法不需要进行高斯积分，因而计算量小．算例表明该方法具有效率高、精度高和稳定性好等优点，并且易于实现．     

随机结构有限元分析的递推求解方法的改进

将随机结构有限元分析的递推求解方法和伽辽金投影方法相结合,提出了求解随机静力响应的改进的递推求解方法。利用随机收敛的非正交多项式展开表示由于材料、外部荷载或构件几何尺寸的随机性导致的结构随机响应。采用递推求解方法得到响应多项式展开的初始系数,并运用定义的数学算子显式地表达出来。然后,通过定义修正系数,应用伽辽金方法对随机力平衡方程在非正交多项式基上进行投影,得到了和响应展开阶次个数相同的确定的有限元方程,并进行求解得到了修正系数。数值算例表明,通过对递推求解方法中响应表达式系数的修正,以很小的计算代价较大地提高了随机响应的计算精度;与基于正交多项式展开的随机有限元方法相比,在精度相当的前提下新方法耗费的计算时间大大降低。     

二维对流扩散方程的高精度全隐式多重网格方法

提出了数值求解二维非定常变系数对流扩散方程的一种时间二阶、空间四阶精度的三层全隐紧致差分格式。为了加快迭代求解隐格式时在每一个时间步上的收敛速度，采用多重网格加速技术，建立了适用于本文高精度金隐紧致格式的多重网格算法。数值实验结果验证了本文方法的精确性、稳定性和对高网格雷诺数问题的强适应性。     

基于非结构化同位网格的SIMPLE算法

通过基于非结构化网格的有限体积法对二维稳态Navier—Stokes方程进行了数值求解。其中对流项采用延迟修正的二阶格式进行离散；扩散项的离散采用二阶中心差分格式；对于压力-速度耦合利用SIMPLE算法进行处理；计算节点的布置采用同位网格技术，界面流速通过动量插值确定。本文对方腔驱动流、倾斜腔驱动流和圆柱外部绕流问题进行了计算，讨论了非结构化同位网格有限体积法在实现SIMPLE算法时，迭代次数与欠松弛系数的关系、不同网格情况的收敛性、同结构化网格的对比以及流场尾迹结构。通过和以往结果比较可知，本文的方法是准确和可信的。     

完全变换法在无网格伽辽金方法中的应用

由于移动最小二乘形函数一般不具有常规有限元或边界元形函数所具有的插值特征，本质边界条件的处理成为无网格伽辽金法实施中的一个难点。本文通过建立节点位移和广义位移之间的关系对移动最小二乘形函数进行修正，给出了修正的移动最小二乘形函数；以二维问题为例，对完全交换法在无网格伽辽金方法中的应用进行了研究，实现了本质边界条件在节点处的精确施加。数值计算结果表明该方法不仅简单合理，而且具有较高的精度、收敛性和稳定性。     

棒材通过锥形模静液挤压成形的无网格法分析

建立了一种基于初始构形及有限变形的粘塑性弹性本构关系，并由空间描述的Galerkin能量弱变分原理，经一致转换得一整体拉格朗日方程描述下的动量平衡方程．同时经线性化处理给出了显示中心差分法求解格式，以棒材通过锥形模的静液挤压成形为例进行了全面的EFG法数值模拟，从而证明了有限变形粘塑性EFG法对实际成形工艺分析、优化及设计的有效性。     

Augmented upwind numerical schemes for the groundwater transport advection‐dispersion equation with local operators

When solute transport is advection‐dominated, the advection‐dispersion equation approximates to a hyperbolic‐type partial differential equation, and finite difference and finite element numerical approximation methods become prone to artificial oscillations. The upwind scheme serves to correct these responses to produce a more realistic solution. The upwind scheme is reviewed and then applied to the advection‐dispersion equation with local operators for the first‐order upwinding numerical approximation scheme. The traditional explicit and implicit schemes, as well as the Crank‐Nicolson scheme, are developed and analyzed for numerical stability to form a comparison base. Two new numerical approximation schemes are then proposed, namely, upwind–Crank‐Nicolson scheme, where only for the advection term is applied, and weighted upwind‐downwind scheme. These newly developed schemes are analyzed for numerical stability and compared to the traditional schemes. It was found that an upwind–Crank‐Nicolson scheme is appropriate if the Crank‐Nicolson scheme is only applied to the advection term of the advection‐dispersion equation. Furthermore, the proposed explicit weighted upwind‐downwind finite difference numerical scheme is an improvement on the traditional explicit first‐order upwind scheme, whereas the implicit weighted first‐order upwind‐downwind finite difference numerical scheme is stable under all assumptions when the appropriate weighting factor (θ) is assigned.     

Numerical solutions to regularized long wave equation based on mixed covolume method

The mixed covolume method for the regularized long wave equation is developed and studied. By introducing a transfer operator γh , which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.     

A stabilised characteristic finite element method for transient Navier–Stokes equations

In this work, we consider a stabilised characteristic finite element method for the time-dependent Navier–Stokes equations based on the lowest equal-order finite element pairs. The diffusion term in these equations is discretised by using finite element method, the temporal differentiation and advection terms are treated by characteristic schemes. Unconditionally stable results and error estimates of optimal order for the velocity and pressure are established. Finally, some numerical results are provided to verify the performance of this method.     

Optimised composite numerical schemes in 2‐D for hyperbolic conservation laws

One of the techniques available for optimising parameters that regulate dispersion and dissipation effects in finite difference schemes is the concept of minimised integrated exponential error for low dispersion and low dissipation. In this paper, we work essentially with the two‐dimensional (2D) Corrected Lax–Friedrichs and Lax–Friedrichs schemes applied to the 2D scalar advection equation. We examine the shock‐capturing properties of these two numerical schemes, and observe that these methods are quite effective from the point of being able to control computational noise and having a large range of stability. To improve the shock‐capturing efficiency of these two methods, we derive composite methods using the idea of predictor/corrector or a linear combination of the two schemes. The optimal cfl number for some of these composite schemes are computed. Some numerical experiments are carried out in two dimensions such as cylindrical explosion, shock‐focusing, dam‐break and Riemann gas dynamics tests. The modified equations of some of the composite schemes when applied to the 2D scalar advection equation are obtained. We also perform some convergence tests to obtain the order of accuracy and show that better results in terms of shock‐capturing property are obtained when the optimal cfl obtained using minimised integrated exponential error for low dispersion and low dissipation is used. Copyright © 2011 John Wiley & Sons, Ltd.     

On dispersion-controlled principles for non-oscillatory shock-capturing schemes

The role of dispersions in the numerical solutions of hydrodynamic equation systems has been realized for long time. It is only during the last two decades that extensive studies on the dispersion-controlled dissipative (DCD) schemes were reported. The studies have demonstrated that this kind of the schemes is distinct from conventional dissipation-based schemes in which the dispersion term of the modified equation is not considered in scheme construction to avoid nonphysical oscillation occurring in shock wave simulations. The principle of the dispersion controlled aims at removing nonphysical oscillations by making use of dispersion characteristics instead of adding artificial viscosity to dissipate the oscillation as the conventional schemes do. Research progresses on the dispersioncontrolled principles are reviewed in this paper, including the exploration of the role of dispersions in numerical simulations, the development of the dispersion-controlled principles, efforts devoted to high-order dispersion-controlled dissipative schemes, the extension to both the finite volume and the finite element methods, scheme verification and solution validation, and comments on several aspects of the schemes from author‘s viewpoint.     

Dispersion error reduction for acoustic problems using the smoothed finite element method (SFEM)

The smoothed finite element method (SFEM), which was recently introduced for solving the mechanics and acoustic problems, uses the gradient smoothing technique to operate over the cell‐based smoothing domains. On the basis of the previous work, this paper reports a detailed analysis on the numerical dispersion error in solving two‐dimensional acoustic problems governed by the Helmholtz equation using the SFEM, in comparison with the standard finite element method. Owing to the proper softening effects provided naturally by the cell‐based gradient smoothing operations, the SFEM model behaves much softer than the standard finite element method model. Therefore, the SFEM can significantly reduce the dispersion error in the numerical solution. Results of both theoretical and numerical experiments will support these important findings. It is shown clearly that the SFEM suits ideally well for solving acoustic problems, because of the crucial effectiveness in reducing the dispersion error. Copyright © 2015 John Wiley & Sons, Ltd.     

Impact of truncation error and numerical scheme on the simulation of the early time growth of viscous fingering

The truncation error associated with different numerical schemes (first order finite volume, second order finite difference, control volume finite element) and meshes (fixed Cartesian, fixed structured triangular, fixed unstructured triangular and dynamically adapting unstructured triangular) is quantified in terms of apparent longitudinal and transverse diffusivity in tracer displacements and in terms of the early time growth rate of immiscible viscous fingers. The change in apparent numerical longitudinal diffusivity with element size agrees well with the predictions of Taylor series analysis of truncation error but the apparent, numerical transverse diffusivity is much lower than the longitudinal diffusivity in all cases. Truncation error reduces the growth rate of immiscible viscous fingers for wavenumbers greater than 1 in all cases but does not affect the growth rate of higher wavenumber fingers as much as would be seen if capillary pressure were present. The dynamically adapting mesh in the control volume finite element model gave similar levels of truncation error to much more computationally intensive fine resolution fixed meshes, confirming that these approaches have the potential to significantly reduce the computational effort required to model viscous fingering.     

A Stokes model with cavitation for the numerical simulation of hydrodynamic lubrication

We present a cavitation model based on the Stokes equation and formulate adaptive finite element methods for its numerical solution. A posteriori error estimates and adaptive algorithms are derived, and numerical examples illustrating the theory are supplied, in particular with comparison to the simplified Reynolds model of lubrication. Copyright © 2010 John Wiley & Sons, Ltd.     

Anisotropic rectangular nonconforming finite element analysis for Sobolev equations

An anisotropic rectangular nonconforming finite element method for solving the Sobolev equations is discussed under semi-discrete and full discrete schemes. The corresponding optimal convergence error estimates and superclose property are derived, which are the same as the traditional conforming finite elements. Furthermore, the global superconvergence is obtained using a post-processing technique. The numerical results show the validity of the theoretical analysis.     

MIXED FINITE ELEMENT METHODS FOR THE SHALLOW WATER EQUATIONS INCLUDING CURRENT AND SILT SEDIMENTATION (Ⅰ)-THE CONTINUOUS-TIME CASE

An initial-boundary value problem for shallow equation system consisting of water dynamics equations, silt transport equation, the equation of bottom topography change, and of some boundary and initial conditions is studied, the existence of its generalized solution and semidiscrete mixed finite element (MFE) solution was discussed, and the error estimates of the semidiscrete MFE solution was derived. The error estimates are optimal.