Adaptive methodology for the meshless Galerkin boundary node method

The Galerkin boundary node method (GBNM) is a boundary-type meshless method that combines a variational form of boundary integral formulations for governing equations with the moving least-squares approximations for generation of the trial and test functions. In this paper, a posteriori error estimate and an effective adaptive h-refinement procedure are developed in conjunction with the GBNM. The error estimator is based on the difference between numerical solutions obtained using two successive nodal arrangements. The reliability and efficiency of this error estimator and the convergence of this adaptive meshless scheme are verified theoretically. Numerical examples are also given to show the efficiency of the adaptive methodology.     

Three-dimensional vibration of cylindrical shell panels – solution by continuum and discrete approaches

 This paper presents a three-dimensional elasticity solution to the free vibration problem of thick cylindrical shell panels of rectangular planform. The natural frequencies and corresponding mode shapes were obtained for thick cylindrical shell panels and detailed parametric investigations were carried out. Comparisons were also made with corresponding finite element simulation results. To validate the accuracy of the results as well as the stability of the present methodology, comprehensive convergence studies were also carried out. Further comparisons of present results were made with existing experimental and numerical results (classical, first-order and higher-order shell theories) available in open literature and the validity and range of applicability of the various shell theories examined.     

Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory

A two-dimensional (2D) higher-order deformation theory is presented for vibration and buckling problems of circular cylindrical shells made of functionally graded materials (FGMs). The modulus of elasticity of functionally graded (FG) shells is assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal deformations, and rotatory inertia is derived through Hamilton’s principle. Several sets of truncated Mth order approximate theories are applied to solve the eigenvalue problems of simply supported FG circular cylindrical shells. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency for the fundamental mode r=s=1 are examined in detail. A comparison of the present natural frequencies of isotropic and FG shells is also made with previously published results. Critical buckling stresses of simply supported FG circular cylindrical shells subjected to axial stress are also obtained and a relation between the buckling stress and natural frequency is presented. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses are calculated by integrating the three-dimensional (3D) equations of motion in the thickness direction satisfying the stress boundary conditions at the outer and inner surfaces. The 2D higher-order deformation theory has an advantage in the analysis of vibration and buckling problems of FG circular cylindrical shells.     

On error estimator and adaptivity in the meshless Galerkin boundary node method

In this article, a simple a posteriori error estimator and an effective adaptive refinement process for the meshless Galerkin boundary node method (GBNM) are presented. The error estimator is formulated by the difference between the GBNM solution itself and its L ²-orthogonal projection. With the help of a localization technique, the error is estimated by easily computable local error indicators and hence an adaptive algorithm for h-adaptivity is formulated. The convergence of this adaptive algorithm is verified theoretically in Sobolev spaces. Numerical examples involving potential and elasticity problems are also provided to illustrate the performance and usefulness of this adaptive meshless method.     

A Galerkin boundary node method for biharmonic problems

A Galerkin boundary node method (GBNM) is developed in this paper for solving biharmonic problems. The GBNM combines an equivalent variational form of boundary integral formulations for governing equations with the moving least-squares approximations for construction of the trial and test functions. In this approach, only a nodal data structure on the boundary of a domain is required. In addition, boundary conditions can be implemented accurately and the system matrices are symmetric. The convergence of this method and numerical examples are given to show the efficiency.     

A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines variational formulations of boundary integral equations with the moving least-squares approximations. This paper presents the mathematical derivation of a posteriori error estimates and adaptive refinement procedures for the GBNM for 3D potential problems. Two types of error estimators are developed in detail. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two successive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the GBNM solution itself and its L²-orthogonal projection. The reliability and efficiency of both types of error estimators is established. That is, these error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization technique is introduced to accommodate the non-local property of integral operators for the needed local and computable a posteriori error indicators. Convergence analysis results of corresponding adaptive meshless procedures are also given. Numerical examples with high singularities illustrate the theoretical results and show that the proposed adaptive procedures are simple, effective and efficient.     

Comparison of the Luus–Jaakola optimization procedure and the genetic algorithm

The success of both genetic algorithms (GA) and the Luus–Jaakola (LJ) optimization procedure in engineering optimization and the desire for efficient optimization methods arising from practical experience make the comparison of these two methods necessary. The GA and the LJ optimization procedure are compared in terms of convergence speed and reliability in obtaining the global optimum. Instead of using the number of function evaluations, this study uses computation time for comparison of convergence speed, which is more precise. Although for some problems, such as parameter estimation for the catalytic cracking process of gas oil, both GA and LJ converge to the optimum rapidly and show high reliability; in most cases, the LJ optimization procedure was found to be faster than GA and exhibited higher reliability in obtaining the global optimum. Furthermore, the LJ optimization procedure is easier to program.     

The meshless Galerkin boundary node method for Stokes problems in three dimensions

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines an equivalent variational formulation of boundary integral equations for governing equations and the moving least‐squares (MLS) approximations for generating the trial and test functions. In this approach, boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Besides, the resulting formulation inherits the symmetry and positive definiteness of the variational problems. The GBNM is developed in this paper for solving three‐dimensional stationary incompressible Stokes flows in primitive variables. The numerical scheme is based on variational formulations for the first‐kind integral equations, which are valid for both interior and exterior problems simultaneously. A rigorous error analysis and convergence study of the method for both the velocity and the pressure is presented in Sobolev spaces. The capability of the method is also illustrated and assessed through some selected numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.     

Nonlinear bending and postbuckling of FGM cylindrical panels subjected to combined loadings and resting on elastic foundations in thermal environments

A nonlinear analysis is presented for FGM cylindrical panels resting on elastic foundations subjected to the combined actions of uniform lateral pressure and compressive edge loads in thermal environments. The two cases of postbuckling of initially pressurized FGM cylindrical panels and of nonlinear bending of initially compressed cylindrical panels are considered. Heat conduction and temperature-dependent material properties are both taken into account. Material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction based on Mori-Tanaka micromechanics model. The formulations are based on a higher order shear deformation theory and von Kármán strain displacement relationships. The panel-foundation interaction and thermal effects are also included. The governing equations are solved by a singular perturbation technique along with a two-step perturbation approach. The numerical illustrations concern the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels with two constituent materials resting on Pasternak elastic foundations. The effects of volume fraction index, temperature variation, foundation stiffness as well as initial stress on the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels are discussed in detail.     

Dynamic characteristics of cylindrical hybrid panels containing viscoelastic layer based on layerwise mechanics

The viscoelastic damping model of the cylindrical hybrid panels with co-cured, free and constrained layers has been developed and investigated by using the refined finite element method based on the layerwise shell theory. The transverse shear and normal strains and the curved geometry are exactly taken into account in the present layerwise shell model, which can depict the zig-zag in-plane and out-of-plane displacements. The damped natural frequencies, modal loss factors and frequency response functions of cylindrical viscoelastic hybrid panels are compared with those of the base composite panel without a viscoelastic layer. The difference in the free vibration and damping of the thin and thick composite laminates and the viscoelastic sandwiched beam between full and partial layerwise theories is verified by comparison with the published results. Various damping characteristics of cylindrical hybrid panels with free viscoelastic layer, constrained layer damping, and co-cured sandwich laminates are investigated. Present results show that the full layerwise damping model accurately predicted the vibration and damping of the cylindrical hybrid panels with viscoelastic layers.     

Three-dimensional free vibration of functionally graded fiber orientation and volume fraction cylindrical panels

Elasticity solution for free vibrations analysis of functionally graded fiber orientation and volume fraction cylindrical panel is presented, using differential quadrature method. The orthotropic panel is simply supported at the edges and assumed to have arbitrary variations of fiber orientation and volume fraction in the radial direction. Suitable displacement functions that identically satisfy the simply supported boundary conditions are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by differential quadrature method to obtain the natural frequencies. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. Numerical results are presented for an orthotropic cylindrical panel with arbitrary variations of fiber orientation and volume fraction in the shell’s thickness and compared with discrete laminates composite panels. The interesting and new results show that normalized natural frequency of the functionally graded fiber orientation cylindrical panel is smaller than that of a discrete laminate composite panel and close to that of a 4-layer. In contrast, the normalized natural frequency of a functionally graded fiber volume fractions is larger than that of a discrete laminated and close to that of a 2-layer.     

A semi-analytical solution for stress analysis of moderately thick laminated cylindrical panels with various boundary conditions

Stress analysis of moderately thick laminated cylindrical panels with different loading and boundary conditions is presented. Boundary conditions include clamped, simply supported and free while uniform and sinusoidal distributed loadings are considered. Assuming effects of shear deformation and initial curvature, governing equations of the problem are derived. The governing partial differential equations (PDEs) in terms of three displacement components, two rotations and ten stress resultants include a system of 15 first order PDEs. Application of the extended Kantorovich method (EKM) to the governing equations yields to a double set of algebric-differential equations in terms of x and θ. The resulted systems are then solved iteratively with very fast convergence. It is demonstrated that the method converges rapidly independent of initial guess functions. Comparisons of the EKM predictions with other analytical and FEM analyses are in close agreement. More results for panels with particular boundary conditions are presented for future studies.     

遗传算法的改进策略及其在桥梁抗震优化设计中的应用效果

本文论述了采用遗传算法进行结构优化设计时遇到的诸如计算量大、早熟收敛和边界探索不足等棘手问题，提出了三个解决对策，编制了计算程序，其在桥梁抗震代化设计中的应用效果表明，改进后的遗传算法不但提高了计算速度，而且在尽可能短的时间内找到最好的优化解。     

Three-dimensional piezoelasticity solution for dynamics of cross-ply cylindrical shells integrated with piezoelectric fiber reinforced composite actuators and sensors

A benchmark three-dimensional (3D) exact piezoelasticity solution is presented for free vibration and steady state forced response of simply supported smart cross-ply circular cylindrical shells of revolutions and panels integrated with surface-bonded or embedded monolithic piezoelectric or piezoelectric fiber reinforced composite (PFRC) layers. The effective properties of PFRC laminas for the 3D case are obtained based on a fully coupled iso-field model. The governing partial differential equations are reduced to ordinary differential equations in the thickness coordinate by expanding all entities for each layer in double Fourier series in span coordinates, which identically satisfy the boundary conditions at the simply-supported ends. These equations with variable coefficients are solved using the modified Frobenius method, wherein the solution is constructed as a product of an exponential function and a power series. The unknown constants of the general solution are finally obtained by employing the transfer matrix method across the layers. Results for natural frequencies and the forced response are presented for single layer piezoelectric and multilayered hybrid composite and sandwich shells of revolution and shell panels integrated with monolithic piezoelectric and PFRC actuator/sensor layers. The present benchmark solution would help assess 2D shell theories for dynamic response of hybrid cylindrical shells.     

基于直接搜索方法的层合板多目标优化设计

直接多目标搜索方法(DMS)是一种不需要计算梯度信息并且能实现全局收敛的多目标优化方法。基于直接多目标搜索方法,以简支层合板铺设角度为设计变量,基频和声功率为目标函数进行层合板结构振动与声多目标优化。分别以4层、8层复合材料层合板为例,用DMS方法对其优化设计,并与传统的遗传算法(GA)对比。结果表明,对于4层复合材料层合板,DMS方法比GA方法优化速度快,且能得到全局最优解;对于8层复合材料层合板,用DMS方法比4层板优化所需时间多,但相比GA方法,DMS方法优化更快。     

An efficient method for nonlinear analysis of layered shells

Finite element formulation based on explicit through-thickness integration scheme assumes importance when applied to multilayered shells, as it is numerically accurate and computationally efficient. Explicit integration becomes possible on assuming the variation of the inverse Jacobian through the thickness. The element stiffness matrices are discussed for (i) large rotation, and (ii) small rotation. Relative efficiencies of the explicit through-thickness integration schemes are compared with that of the conventional formulation involving numerical integration in three directions in each layer and summation over the layers. The small rotation formulation assuming linear variation of the Jacobian inverse across the thickness and based on further approximation regarding certain submatrices is seen to be computationally efficient. The geometric nonlinear behaviours of laminated composite cylindrical panels subjected to external pressure are discussed. The parameters considered are: number of layers, symmetric/antisymmetric, cross-ply/angleply, boundary conditions and central angle. The strength of shallow panels with longitudinal edges hinged and curved edges free is controlled by the limit point load, while for deep panels it is controlled by the bifurcation load. The boundary conditions have significant influence on load carrying capacities. A list of symbols is given at the end of the paper     

基于解析解和边界元解的圆柱壳声辐射对比研究

针对工程广泛应用的圆柱壳结构,首次运用解析法探讨了边界元软件Sysnoise两种加载方式—基于单元(Element)加载和基于节点(Node)加载—对声辐射影响,以及两种加载方式在网格加密情况下的收敛性问题。得到一些对工程计算的有益结论：① 基于单元(Element)加载方式得到的结果要好于基于节点(Node)加载方式;② 基于单元(Element)加载方式网格加密时收敛到解析解,而基于节点(Node)加载方式网格加密后,不一定得到收敛结果。     

Post-buckling finite element analysis of composite cylindrical panels in axial compression

The post-buckling behaviour of composite cylindrical panels in axial compression is investigated using the nonlinear finite element theory. A branch switching algorithm with an ‘eigenmode injection’ is discussed. The effect of the reinforcement angle on the buckling load, as well as the post-buckling behaviour of the panels, are analysed. Examples are considered for separated and closely spaced bifurcation points.

By varying the reinforcement angle of the panels from 0 to 90°, two kinds of buckling modes are discovered for asymmetric and unstable symmetric lowest points of bifurcation. Local minima for the secondary branches of geometrically perfect panels are no less than 15% of their buckling loads, while values of the buckling loads alter more than twice in this region.

A 16-node degenerated shell element with full integration scheme is used.     

A genetic algorithm approach to single and multiobjective structural optimization with discrete–continuous variables

A search procedure with a philosophical basis in molecular biology is adapted for solving single and multiobjective structural optimization problems. This procedure, known as a genetic algorithm (GA). utilizes a blending of the principles of natural genetics and natural selection. A lack of dependence on the gradient information makes GAs less susceptible to pitfalls of convergence to a local optimum. To model the multiple objective functions in the problem formulation, a co-operative game theoretic approach is proposed. Examples dealing with single and multiobjective geometrical design of structures with discrete–continuous design variables, and using artificial genetic search are presented. Simulation results indicate that GAs converge to optimum solutions by searching only a small fraction of the solution space. The optimum solutions obtained using GAs compare favourably with optimum solutions obtained using gradient-based search techniques. The results indicate that the efficiency and power of GAs can be effectively utilized to solve a broad spectrum of design optimization problems with discrete and continuous variables with similar efficiency.     

Optimum structural design of CFRP isogrid cylindrical shell using genetic algorithm

This paper describes an optimum structural design of a CFRP isogrid cylindrical shell using a genetic algorithm (GA). When the CFRP isogrid cylindrical shell receives a prescribed uniaxial compressive load, an objective is to minimize weight of the CFRP isogrid cylindrical shell subjected to the constraint conditions of no buckling and no material failure. The buckling and material failure loads were approximated by a response surface method combined with partitioning of design spaces and these approximated values were used in the process of GA instead of FEM calculations in order to reduce the computational time. Furthermore, the differences from the constraint conditions of the linear or the non-linear (local) buckling loads were also calculated and their results were compared with each other.