Comparison of numerical schemes for solving the advection equation

We report on the dispersion and dissipation properties of numerical schemes aimed at solving the one-dimensional advection equation. The study is based on the consistency error, which is explicitly calculated for various standard finite-difference schemes. The oscillation and damping features of the numerical solutions are shown to be explained via a generalized Airy-like function. In the specific case of the advection of a step function, the solutions of the equivalent equations are systematically calculated and shown to recover the numerical solutions. A particular emphasis is put on one third-order accurate scheme, which involves a weak smearing of the step.     

Third-order accurate finite-volume method on a triangular grid

A third-order accurate finite-volume method on triangular grids is proposed for the numerical solution of conservation law systems. The method is described in detail as applied to the advection equation. The accuracy of its numerical solutions obtained on grids of various degrees of detail is compared. The viscous flow over a plate and the unsteady flow around a cylinder are solved. For comparison purposes, the latter problem is also solved using third- and fifth-order accurate compact approximations.     

不同理论下圆板特征值之间的解析关系

基于经典板理论(CPT)、一阶剪切变形板理论(FPT)以及Reddy三阶剪切变形板理论(RPT)之间,圆板轴对称特征值问题在数学上的相似性,研究了不同理论之间圆板特征值间的解析关系．将特征值问题的求解转化为代数方程的求解,并导出了不同理论之间圆板特征值的显式精确解析关系．从而,只要已知圆板特征值（临界屈曲载荷和固有频率）的经典结果,便很容易从这些解析关系中得到一阶和三阶理论下圆板特征值的相应结果,这便于工程应用,同时也可检验一阶和三阶理论下板特征值的数值结果的有效性、收敛性以及精确性等问题．     

A new refined simple TSDT-based effective meshfree method for analysis of through-thickness FG plates

In this paper a novel numerical method based on the Moving Kriging (MK) interpolation meshfree method, integrated with a simple higher-order shear deformation plate theory for analysis of static bending, free vibration and buckling of functionally graded (FG) plates is presented. In the proposed technique, the shape functions are built by the Kriging technique which possesses the property of Kronecker delta function which makes it easy to enforce essential boundary conditions. The present formulation is based on a refined simple third-order shear deformation theory (R-STSDT), which is based on four variables and it still accounts for parabolic distribution of the transverse shearing strains and stresses through the thickness of the plate present in the original simple third-order shear deformation theory (STSDT). In this theory, instead of assuming a specific distribution for the displacement field, the theory of elasticity is used for obtaining the kinematics of the plate deformation. We first propose the formulation, and then several numerical examples are provided to show the merits of the proposed approach.     

A General Darboux-Type Boundary Value Problem in Curvilinear Domains with Corners for a Third-Order Equation with Dominated Lower-Order Terms

We study solvability of the Darboux-type boundary value problem for a third-order linear partial differential equation with dominated lower-order terms. We indicate function spaces in which the problem is uniquely solvable and Hausdorff normally solvable. In the second case, the corresponding homogeneous problem is shown to have infinitely many linearly independent solutions.     

Finite-Difference Method for a System of Third-Order Boundary-Value Problems

In this paper, we develop a two-stage numerical method for computing the approximate solutions of third-order boundary-value problems associated with odd-order obstacle problems. We show that the present method is of order two. A numerical example is presented to illustrate the applicability of the new method. A comparison is also given with previously known results.     

Marangoni-driven liquid films rising out of a meniscus onto a nearly-horizontal substrate

We revisit the situation of a thin liquid film driven up an inclined substrate by a thermally induced Marangoni shear stress against the opposing parallel component of gravity. In contrast to previous studies, we focus here on the meniscus region, in a case where the substrate is nearly horizontal. Our numerical simulations show that the time-dependent lubrication model for the film profile can reach a steady state in the meniscus region that is unlike the monotonic solutions investigated earlier. A systematic investigation of the steady states of the lubrication model is carried out by studying the phase space of the corresponding third-order ODE system. We find a rich structure of the phase space including multiple non-monotonic solutions with the same far-field film thickness. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stress and fracture analysis in delaminated orthotropic composite plates using third-order shear deformation theory

The third-order shear deformable plate theory is applied in this work to calculate the stresses and energy release rates in delaminated orthotropic composite plates with straight crack front. The delaminated parts are modeled by the general third-order plate theory, while a double-plate model with interface constraint is developed for the uncracked portion of the plate. The governing equations of the uncracked part are formulated by considering the equilibrium and the displacement continuity along the interface. As an example, a simply-supported delaminated orthotropic plate subjected to a point force is solved adopting Lévy plate formulation and the state-space approach. The mode-II and mode-III energy release rate distributions along the crack front were calculated by the J-integral. To verify the analytical results the 3D finite element model of the plate was constructed and the energy release rates were calculated by the virtual crack-closure technique. A previous second-order plate theory solution was also utilized in the course of the comparison. The results indicate a good agreement between analysis and numerical computation and that third-order theory is better in some cases than the second-order approximation.     

Compact Difference Scheme with High Accuracy for One-Dimensional Unsteady Quasi-Linear Biharmonic Problem of Second Kind: Application to Physical Problems

The present paper uses a new two-level implicit difference formula for the numerical study of one-dimensional unsteady biharmonic equation with appropriate initial and boundary conditions. The proposed difference scheme is second-order accurate in time and third-order accurate in space on non-uniform grid and in case of uniform mesh, it is of order two in time and four in space. The approximate solutions are computed without using any transformation and linearization. The simplicity of the proposed scheme lies in its three-point spatial discretization that yields block tri-diagonal matrix structure without the use of any fictitious nodes for handling the boundary conditions. The proposed scheme is directly applicable to singular problems, which is the main utility of our work. The method is shown to be unconditionally stable for model linear problem for uniform mesh. The efficacy of the proposed approach has been tested on several physical problems, including the complex fourth-order nonlinear equations like Kuramoto–Sivashinsky equation and extended Fisher–Kolmogorov equation, where comparison is done with some earlier work. It is clear from numerical experiments that the obtained results are not only in good agreement with the exact solutions but also competent with the solutions derived in earlier research studies.     

Simulating thin plate bending problems by a family of two-parameter homogenization functions

This study develops a simple and effective numerical technique, which aims to accurately and quickly address thin plate bending problems. Based on the given boundary conditions, the thin plate homogenization function is constructed and a family of two-parameter homogenization functions are derived. Then, the superposition of homogenization functions method for the thin plate, the clamped plate, and the simply supported plate is obtained, which is meshless without numerical integration and iteration with the merits of easy-to-program and easy-to-implement. Six numerical experiments are employed to verify the effectiveness, accuracy and convergence of the proposed novel strategy. The proposed method is evaluated by the comparisons with the analytical solutions and the referenced solutions. It can be observed that the proposed method is quite accurate for the thin plate, the clamped plate, and the simply supported plate problems.     

有限深两层流中内孤立波的高阶解*

本文研究有限水深两层流中孤立波的三阶近似理论,并考虑了自由表面对孤立波的影响,运用坐标变形方法得到了三阶内孤立波的发展方程,求得波速的解析表达式。对方程进行了数值计算,得到了几种参数下三阶解曲线,指出自由表面对波型和波速的影响是二阶的。计算表明三阶解对一阶、二阶解有明显的改进,使其更加接近试验结果。     

Convective heat transfer in a conducting fluid over a permeable stretching surface with suction and internal heat generation/absorption

We consider a convective flow in a porous medium of an incompressible viscous conducting fluid impinging on a permeable stretching surface with suction, and internal heat generation/absorption. Using a similarity transformation the governing equations of the problem are reduced to a coupled third-order nonlinear ordinary differential equations. We first examine a number of special cases for which we may obtain exact solutions. We then obtain analytical solutions (by the Homotopy Analysis Method) and numerical solutions (by a boundary value problem solver), in order to further study the behavior of the nonlinear differential equations, for various values of the physical parameters. Our numerical solutions are shown to agree with the available results in the literature. We then employ the numerical results to bring out the effects of the suction parameter, heat source/sink parameter, stretching parameter, porosity parameter, the Prandtl number and the free convection parameter on the flow and heat transfer characteristics. In the absence of suction and free convection, our findings are in agreement with the corresponding numerical results of Attia [H.A. Attia, On the effectiveness of porosity on stagnation point flow towards a stretching surface with heat generation, Comput. Mater. Sci. 38 (2007) 741-745].     

A multiple-scales solution of the undular hydraulic jump problem

A non-linear third-order ODE governing the free surface of an undular hydraulic jump is solved by means of a multiple-scales analysis. The resulting solutions are compared with direct numerical solutions of the ODE, and good agreement is found. Furthermore, new flow features, which are not evident in the numerical solutions, are revealed by the multiple-scales solutions. The results of the analysis are also compared with numerical solutions of the full equations of motion and with experimental data, and satisfactory agreement is obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

变厚度圆板大挠度理论的摄动法

本文用对两个小参数的摄动法,对于轴对称圆薄板大挠度问题,在板厚按指数规律变化、载荷为均布的情况下,求出了三级摄动解。所得摄动解在特殊情况下与精确解的比较表明结果是较为理想的。     

Mid-frequency dynamic characteristics prediction of thin plate based on B-spline wavelet on interval finite element method

Due to low computing efficiency and dispersion errors, Traditional Finite Element Methods (TFEMs) based on general polynomials cannot provide efficient dynamic solutions within mid-frequency domain which is the gap between low and high frequency domain. It is also defined as mid-frequency problem in the field of sound and vibration analysis. To solve this problem, it is essential to overcome these two disadvantages simultaneously based on much better computing efficiency and numerical stability. Fortunately, due to the multi-scale/multi-resolution features, the c₁ type Wavelet Finite Element Methods (WFEMs) own much better computing efficiency and numerical stability. Therefore, WFEMs will be introduced for dealing with the low computing efficiency and dispersion errors and solving the mid-frequency problem based on multi-element analysis. But, due to the complex nodes numbering and Degree of Freedoms (DOFs) numbering, the c₁ type WFEMs combined with existing assembling formulas cannot provide efficient solutions by multi-element analysis any more. Therefore, this paper mainly consists of two parts of research work. On the one hand, the proper assembling formulas are derived detailedly based on c₁ type WFEMs. On the other hand, the method combining c₁ type B-spline wavelet thin plate element with the newly derived assembling formulas is proposed for predicting dynamic characteristics and solving mid-frequency problem related to thin plate structures. The numerical study shows that both computing efficiency and numerical stability of the proposed method are much better than TFEMs’. Furthermore, the proposed method's prediction ability can break through the limitation of TFEMs’ highest computing accuracy. In addition, the proposed method is verified by experimental study for predicting acceleration Frequency Response Functions (FRFs) of thin plate within 5 Hz–1000 Hz, and the experimental results indicate that the proposed method provides the potential to solve mid-frequency problem related to thin plate structures.     

Positive solutions of discrete third-order three-point right focal boundary value problems

We consider linear and nonlinear boundary value problems for third-order difference equations with three-point right focal boundary conditions. In the linear case, the existence of positive solutions corresponding to the first eigenvalue of the problem is established and an interval estimate for the first eigenvalue is obtained. In the nonlinear case, sufficient conditions for the existence and nonexistence of positive solutions are obtained. An example is included at the end of the paper to illustrate the sharpness of our results.     

Generalizations of a result of Christensen on renewal sequences and linear recurrences

This note extends some recent work of Christensen on simple closed form solutions for linear recurrences, employing connections with renewal sequences. We obtain results for linear recursive sequences defined via monotone coefficients, and provide a broader result in the third-order case. A related open question is posed.     

Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third-order shear deformation plate theory

In this paper, exact closed-form solutions in explicit forms are presented for transverse vibration analysis of rectangular thick plates having two opposite edges hard simply supported (i.e., Lévy-type rectangular plates) based on the Reddy’s third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. Hamilton’s principle is used to derive the equations of motion and natural boundary conditions of the plate. Several comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate accuracy of the present new formulation. Comprehensive benchmark results for natural frequencies of rectangular plates with different combinations of boundary conditions are tabulated in dimensionless form for various values of aspect ratios and thickness to length ratios. A set of three-dimensional (3-D) vibration mode shapes along with their corresponding contour plots are plotted by using exact transverse displacements of Lévy-type rectangular Reddy plates. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

Three-dimensional problems of steady vibrations of isotropic plates

Using the symbolic method of homogeneous solutions, we study problems of steady vibrations of isotropic plates. On the planar faces of the plate we state various kinds of homogeneous boundary conditions. We obtain the homogeneous solutions of the equations of motion and construct the dispersion equations. We carry out numerical analyses of the dispersion equations for a plate with clamped and planar faces. Four figures. Bibliography: 8 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 13–19.     

Global existence for the full von Kármán system

We consider here the full system of dynamic von Kármán equations, taking into account the in-plane acceleration terms, which is a model for the vibrations of a nonlinear elastic plate. We prove global existence and uniqueness of strong solutions for this system with various boundary conditions possibly including feedback terms which are useful for stabilization purposes.