Mesh Discretization of the Vector Relations of Shell Theory in a Curvilinear Coordinate System

Mesh methods for discretization of the differential vector relations are generalized as applied to problems of shell theory. In the finite-difference method, covariant derivatives are replaced by vector differences, which are then projected on the vectors of a local basis. In the finite-element method, vector functions are approximated by a Taylor series with tensor coefficients. It is shown that such schemes satisfy the condition of rigid displacement for a deformable body, which improves considerably the convergence of the solution. The proposed schemes, which are sensitive to approximation uncertainties, were tested by solving problems on deformation of shells     

A STUDY ON THE WEIGHT FUNCTION OF THE MOVING LEAST SQUARE APPROXIMATION IN THE LOCAL BOUNDARY INTEGRAL EQUATION METHOD

The meshless method is a new numerical technique presented in recent years .It uses the moving least square (MLS) approximation as a shape function . The smoothness of the MLS approximation is determined by that of the basic function and of the weight function, and is mainly determined by that of the weight function. Therefore, the weight function greatly affects the accuracy of results obtained. Different kinds of weight functions, such as the spline function, the Gauss function and so on, are proposed recently by many researchers. In the present work, the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method. The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed. Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and a in Gauss and exponential weight functions are in the range of reasonable values, respectively, and the higher the smoothness of the weight function, the better the features of the solutions.     

Study of the Deformation of Anisotropic Viscoelastic Bodies

A study is made of methods for solving linear viscoelastic problems on the basis of the Volterra concept — representation of irrational functions of integral operators as operator power series (analogues of Taylor series). It is pointed out that these series converge weakly. The results of development and substantiation of a new mathematical method for solution of the above problems are summarized. It is based on representing irrational functions of integral operators by operator continued fractions, which converge well. Solutions to certain linear viscoelastic problems for anisotropic bodies are given     

Galerkin technique based on beam functions in application to the parametric instability of thermal convection in a vertical slot

A Fourier–Galerkin spectral technique for solving coupled higher‐order initial‐boundary value problems is developed. Conjugated systems arising in thermoconvection that involve both equations of fourth and second spatial orders are considered. The set of so‐called beam functions is used as basis together with the harmonic functions. The necessary formulas for expressing each basis system into series with respect to the other are derived. The convergence rate of the spectral solution series is thoroughly investigated and shown to be fifth‐order algebraic for both linear and nonlinear problems. Though algebraic, the fifth‐order rate of convergence is fully adequate for the generic problems under consideration, which makes the new technique a useful tool in numerical approaches to convective problems. An algorithm is created for the implementation of the method and the results are thoroughly tested and verified on different model examples. The spatial and temporal approximation of the scheme is tested. To further validate the scheme, a singular asymptotic expansion is derived for small values of the modulation frequency and amplitude and the numerical and analytic results are found to be in good agreement. The new technique is applied to the G‐jitter flow, and the Floquet stability diagrams are produced. We obtain the expected alternating isochronous and subharmonic branches and find that stable motions are always isochronous while unstable motions can be either isochronous or subharmonic. The numerical investigation also leads to novel conclusions regarding the dependence of the amplitude of the solutions on some of the governing parameters. Copyright © 2008 John Wiley & Sons, Ltd.     

基于单位分解法的无网格数值流形方法

在数值流形方法和单位分解法的基础上，提出了无网格数值流形方法. 无网格数值流形 方法在分析时采用了双重覆盖系统，即数学覆盖和物理覆盖. 数学覆盖提供的节点形成求解 域的有限覆盖和单位分解函数；而物理覆盖描述问题的几何区域及其域内不连续性. 与原有 的数值流形方法相比，无网格数值流形方法的数学覆盖形状更加灵活，可以用一系列节点的 影响域来建立数学覆盖和单位分解函数，具有无网格方法的特性，从而摆脱了传统的数值流 形方法中网格所带来的困难. 与无网格方法相比，由于采用了有限覆盖技术，试函数的构造 不受域内不连续的影响，克服了原有的无网格方法在处理不连续问题时所遇到的困难. 详细推导了无网格数值流形方法的试函数和求解方程，最后给出了算例，验证了该方法的正 确性.     

Analysis of thick plates by local radial basis functions-finite differences method

Although global collocation with radial basis functions proved to be a very accurate means of solving interpolation and partial differential equations problems, ill-conditioned matrices are produced, making the choice of the shape parameter a crucial issue. The use of local numerical schemes, such as finite differences produces better conditioned matrices. For scattered points, a combination of finite differences and radial basis functions avoids the limitation of finite differences to be used on special grids. In this paper, we use a higher-order shear and normal deformation plate theory and a radial basis function—finite difference technique for predicting the static behavior of thick plates. Through numerical experiments on square and L-shaped plates, the accuracy and efficiency of this collocation technique is demonstrated, and the numerical accuracy and convergence are thoughtfully examined. This technique shows great potential to solve large engineering problems without the issue of ill-conditioning.     

一种在网格内部捕捉间断的Walsh函数有限体积方法

传统有限体积或有限元方法假定流动变量在单元内连续,间断仅限于控制体的交界面上,因此它们无法在控制体内部捕捉间断.本文摒弃控制体内流动变量连续的假设,将自身具有间断特点的Walsh基函数应用于有限体积方法,把控制体内的流场变量表示成间断基函数的组合形式.按照Walsh基函数在控制体内引入的间断数目和位置,将控制体单元虚分...     

The dimension split element-free Galerkin method for three-dimensional potential problems

This paper presents the dimension split element-free Galerkin (DSEFG) method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems. For these two-dimensional problems, the improved moving least-squares (IMLS) approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions. The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition. The finite difference method is selected in the splitting direction. For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method. The convergence study and error analysis of the DSEFG method are presented. The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than the IEFG method.     

The Method of R-Functions in the Solution of Elastic Problems on the Basis of Reissner's Mixed Variational Principle

A method is presented for solving boundary-value elastic problems on the basis of the variational–structural method of R-functions and Reissner's mixed variational principle. A mathematical formulation is given to problems on the deformation of elastic bodies under mixed boundary conditions and bodies interacting with smooth rigid dies. Solutions satisfying all the boundary conditions are proposed. For undetermined components of these solutions, the resolving equations are derived and their properties are studied. A posteriori estimation of numerical solutions is made. As examples, solutions are found to a problem on the stress–strain state of a short cylinder and to a contact problem on a cylinder interacting with a smooth die. A numerical method of solving such problems is analyzed for convergence, and the accuracy of the solutions is estimated.     

Static response analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB

In this paper, based on the second-order Taylor series expansion and the difference of convex functions algorithm for quadratic problems with box constraints(the DCA for QB), a new method is proposed to solve the static response problem of structures with fairly large uncertainties in interval parameters. Although current methods are effective for solving the static response problem of structures with interval parameters with small uncertainties, these methods may fail to estimate the region of the static response of uncertain structures if the uncertainties in the parameters are fairly large. To resolve this problem, first, the general expression of the static response of structures in terms of structural parameters is derived based on the second-order Taylor series expansion. Then the problem of determining the bounds of the static response of uncertain structures is transformed into a series of quadratic problems with box constraints. These quadratic problems with box constraints can be solved using the DCA approach effectively. The numerical examples are given to illustrate the accuracy and the efficiency of the proposed method when comparing with other existing methods.     

基于径向基函数配点法的梁板弯曲问题分析

采用径向基函数配点法分析考虑剪切效应的梁板弯曲问题,该方法利用径向基函数作为近似函数,基于配点法离散方程,通过最小二乘法求解。径向基函数配点法在离散和计算过程中不需要任何形式的网格划分,是一种真正的无网格法;径向基函数可以用一元函数来描述多元函数,存在明显的储存和运算简单的特点;而基于配点法求解不需要积分,提高了计算效率。分析考虑剪切效应的薄梁板问题时,传统的有限元法或无网格法求解均会存在剪切锁闭问题,而径向基函数在全域内存在无限连续性,能够准确地满足Kirchhoff约束条件,因此径向基函数配点法能够消除剪切锁闭现象,而且不会出现应力波动。该方法的优势在于,其不仅易于离散、精度高,而且具有指数收敛率,计算效率高。数值算例验证了上述结论和该方法的稳定性。     

一种适用于强非线性结构力学问题数值求解的修正小波伽辽金方法

论文通过对有限区间上的任一连续函数在边界处采用基于泰勒展开的延拓处理,构造了一种与任意边界条件相协调的改进小波尺度基函数及在此基础上建立了小波逼近格式,由此可有效避免小波逼近在求解微分方程时在边界处的跳跃或抖动问题.在此基础上,结合论文后两位作者提出的广义小波高斯积分法,关于未知函数的任意非线性项的小波展开可以显式地用...     

A semi-analytical method with a system of decoupled ordinary differential equations for three-dimensional elastostatic problems

In this paper, a new semi-analytical method is presented for modeling of three-dimensional (3D) elastostatic problems. For this purpose, the domain boundary of the problem is discretized by specific subparametric elements, in which higher-order Chebyshev mapping functions as well as special shape functions are used. For the shape functions, the property of Kronecker Delta is satisfied for displacement function and its derivatives, simultaneously. Furthermore, the first derivatives of shape functions are assigned to zero at any given node. Employing the weighted residual method and implementing Clenshaw–Curtis quadrature, coefficient matrices of equations’ system are converted into diagonal ones, which results in a set of decoupled ordinary differential equations for solving the whole system. In other words, the governing differential equation for each degree of freedom (DOF) becomes independent from other DOFs of the domain. To evaluate the efficiency and accuracy of the proposed method, which is called Decoupled Scaled Boundary Finite Element Method (DSBFEM), four benchmark problems of 3D elastostatics are examined using a few numbers of DOFs. The numerical results of the DSBFEM present very good agreement with the results of available analytical solutions.     

The improvement of numerical modeling in the solution of incompressible viscous flow problems using finite element method based on spherical Hankel shape functions

In this paper, the finite element method with new spherical Hankel shape functions is developed for simulating 2‐dimensional incompressible viscous fluid problems. In order to approximate the hydrodynamic variables, the finite element method based on new shape functions is reformulated. The governing equations are the Navier‐Stokes equations solved by the finite element method with the classic Lagrange and spherical Hankel shape functions. The new shape functions are derived using the first and second kinds of Bessel functions. In addition, these functions have properties such as piecewise continuity. For the enrichment of Hankel radial basis functions, polynomial terms are added to the functional expansion that only employs spherical Hankel radial basis functions in the approximation. In addition, the participation of spherical Bessel function fields has enhanced the robustness and efficiency of the interpolation. To demonstrate the efficiency and accuracy of these shape functions, 4 benchmark tests in fluid mechanics are considered. Then, the present model results are compared with the classic finite element results and available analytical and numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic finite element method.     

Determination of stress intensity factors by the finite element discretized symplectic method

When rewriting the governing equations in Hamiltonian form, analytical solutions in the form of symplectic series can be obtained by the method of separation of variable satisfying the crack face conditions. In theory, there exists sufficient number of coefficients of the symplectic series to satisfy any outer boundary conditions. In practice, the matrix relating the coefficients to the outer boundary conditions is ill-conditioned unless the boundary is very simple, e.g., circular. In this paper, a new two-level finite element method using the symplectic series as global functions while using the conventional finite element shape functions as local functions is developed. With the available classical finite elements and symplectic series, the main unknowns are no longer the nodal displacements but are the coefficients of the symplectic series. Since the first few coefficients are the stress intensity factors, post-processing is not required. A number of numerical examples as well as convergence studies are given.     

HC轧机辊间接触与辊系变形的Taylor级数多极边界元数学规划解析

分析HC轧机辊间接触分布和辊系弹性变形对于改善辊间压力分布状态,减少轧故褂檬倜案纳瓢逍畏浅Ｖ匾?醯捎诩扑懔亢艽?使用传统数值方法(有限元法或边界元法)分析辊间接触和辊系变形是非常困难的.本文描述了一种基于点-面接触模型的三维弹性接触Taylor级数多极边界元法,给出了数学规划解析方法,适合大规模弹性接触问题的求解....     

Local moving least square‐one‐dimensional integrated radial basis function networks technique for incompressible viscous flows

This paper presents a local moving least square‐one‐dimensional integrated radial basis function networks method for solving incompressible viscous flow problems using stream function‐vorticity formulation. In this method, the partition of unity method is employed as a framework to incorporate the moving least square and one‐dimensional integrated radial basis function networks techniques. The major advantages of the proposed method include the following: (i) a banded sparse system matrix which helps reduce the computational cost; (ii) the Kronecker‐ δ property of the constructed shape function which helps impose the essential boundary condition in an exact manner; and (iii) high accuracy and fast convergence rate owing to the use of integration instead of conventional differentiation to construct the local radial basis function approximations. Several examples including two‐dimensional (2D) Poisson problems, lid‐driven cavity flow and flow past a circular cylinder are considered, and the present results are compared with the exact solutions and numerical results from other methods in the literature to demonstrate the attractiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Plate/shell structure topology optimization of orthotropic material for buckling problem based on independent continuous topological variables

The purpose of the present work is to study the buckling problem with plate/shell topology optimiza-tion of orthotropic material.A model of buckling topology optimization is established based on the independent,con-tinuous, and mapping method, which considers structural mass as objective and buckling critical loads as constraints. Firstly, composite exponential function (CEF) and power function(PF)as filter functions are introduced to recognize the element mass,the element stiffness matrix,and the ele-ment geometric stiffness matrix.The filter functions of the orthotropic material stiffness are deduced. Then these fil-ter functions are put into buckling topology optimization of a differential equation to analyze the design sensitiv-ity.Furthermore,the buckling constraints are approximately expressed as explicit functions with respect to the design vari-ables based on the first-order Taylor expansion.The objective function is standardized based on the second-order Taylor expansion. Therefore,the optimization model is translated into a quadratic program.Finally,the dual sequence quadratic programming(DSQP)algorithm and the global convergence method of moving asymptotes algorithm with two different filter functions(CEF and PF)are applied to solve the opti-mal model.Three numerical results show that DSQP&CEF has the best performance in the view of structural mass and discretion.     

Improved non-singular local boundary integral equation method

When the source nodes are on the global boundary in the implementation of local boundary integral equation method (LBIEM),singularities in the local boundary integrals need to be treated specially. In the current paper,local integral equations are adopted for the nodes inside the domain trod moving least square approximation (MLSA) for the nodes on the global boundary,thus singularities will not occur in the new al- gorithm.At the same time,approximation errors of boundary integrals are reduced significantly.As applications and numerical tests,Laplace equation and Helmholtz equa- tion problems are considered and excellent numerical results are obtained.Furthermore, when solving the Hehnholtz problems,the modified basis functions with wave solutions are adapted to replace the usually-used monomial basis functions.Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.     

Collocated discrete least squares meshless (CDLSM) method for the solution of transient and steady‐state hyperbolic problems

A collocated discrete least squares meshless method for the solution of the transient and steady‐state hyperbolic problems is presented in this paper. The method is based on minimizing the sum of the squared residuals of the governing differential equation at some points chosen in the problem domain as collocation points. The collocation points are generally different from nodal points, which are used to discretize the problem domain. A moving least squares method is employed to construct the shape functions at nodal points. The coefficient matrix is symmetric and positive definite even for non‐symmetric hyperbolic differential equations and can be solved efficiently with iterative methods. The proposed method is a truly meshless method and does not require numerical integration. Advantages of the collocation points are shown to be threefold: First, the collocation points are shown to be responsible for stabilizing the method in particular when problems with shocked solution are attempted. Second, the collocation points are also shown to improve the accuracy of the solution even for problems with smooth solutions. Third, the collocation points are shown to contribute to the efficiency of the method when solving steady‐state problems via faster convergence of the resulting algorithm. The ability of the method and in particular the effect of collocation points are tested against a series of one‐dimensional transient and steady‐state benchmark examples from the literature and the results are presented. A sensitivity analysis is also carried out to investigate the effect of the base polynomials on the accuracy and convergence characteristics of the method in solving steady‐state problems. The results show the ability of the proposed method to accurately solve difficult hyperbolic problems considered. The method is also shown to be particularly stable for problems with shocked solution due to the inherent stabilizing mechanism of the method. Copyright © 2008 John Wiley & Sons, Ltd.