A three‐noded shear‐flexible curved beam element based on coupled displacement field interpolations

An efficient shear‐flexible three‐noded curved beam element is proposed herein. The shear flexibility is based on Timoshenko beam theory and the element has three degrees of freedom, viz., tangential displacement (u), radial displacement (w) and the section‐rotation (θ). A quartic polynomial interpolation for flexural rotation ψ is assumed a priori. Making use of the physical composition of θ in terms of ψ and u, a novel way of deriving the polynomial interpolations for u and w is presented, by solving force‐moment and moment‐shear equilibrium equations simultaneously. The field interpolation for θ is then constructed from that of ψ and u. The procedure leads to high‐order polynomial field interpolations which share some of the generalized degrees of freedom, by means of coefficients involving material and geometric properties of the element. When applied to a straight Euler–Bernoulli beam, all the coupled coefficients vanish and the formulation reduces to classical quintic‐in‐w and quadratic‐in‐u element, with u, w, and ?w/?x as degrees of freedom. The element is totally devoid of membrane and shear locking phenomena. The formulation presents an efficient utilization of the nine generalized degrees of freedom available for the polynomial interpolation of field variables for a three‐noded curved beam element. Numerical examples on static and free vibration analyses demonstrate the efficacy and locking‐free property of the element. Copyright © 2001 John Wiley & Sons, Ltd.     

Four-parameter functionally graded cracked plates of arbitrary shape: A GDQFEM solution for free vibrations

The dynamic behavior of moderately thick FGM plates with geometric discontinuities and arbitrarily curved boundaries is investigated. The Generalized Differential Quadrature Finite Element Method (GDQFEM) is proposed as a numerical approach. The irregular physical domain in Cartesian coordinates is transformed into a regular domain in natural coordinates. Several types of cracked FGM plates are investigated. It appears that GDQFEM is analogous to the well-known Finite Element Method (FEM). With reference to the proposed technique the governing FSDT equations are solved in their strong form and the connections between the elements are imposed with the inter-element compatibility conditions. The results show excellent agreement with other numerical solutions obtained by FEM.     

Fully coupled linear modelling of incompressible free‐surface flow,compressible air and flexible structures

Incompressible free‐surface flow is a common assumption for the modelling of water waves. Connected with the aim to develop very large floating platforms, air chamber supported floating structures have attracted considerable research interest in the past. Such structures are carried by air entrapped in chambers formed by stiff, vertical walls. In order to model these types of structures, the interactions between surface gravity waves and compressible air must be taken into account. If the payload requirements for air chamber supported structures are low enough, the air chambers may be formed by flexible membrane cylinders. In such systems, pressure variations can lead to considerable changes in chamber volume. Therefore, the flexibility of the bounding structures must be taken into account. We present a modelling strategy to tackle the fully coupled problem of compressible gas in a flexible chamber and incompressible free‐surface flow in an unbounded domain. The governing equations and boundary conditions are described and solved by the finite element method. A perfectly matched layer is used to obtain a solution for an unbounded domain. Finally, the numerical implementation is validated by various test cases. Copyright © 2015 John Wiley & Sons, Ltd.     

hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

A high‐order generalized finite element method (GFEM) for non‐planar three‐dimensional crack surfaces is presented. Discontinuous p‐hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non‐planar, non‐smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non‐planar curved crack fronts and the robustness of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

An accurate four‐node shear flexible composite plate element

An efficient and accurate four node shear flexible composite laminated plate element with six degrees of freedom per node, viz. three displacement (u, v, w) along the x‐, y‐and z‐axis, two rotations (θ_x and θ_y) about y‐ and x‐ axis and twist (θ_xy) is proposed in this paper. A coupled displacement field is derived using moment–shear equilibrium and in‐plane equilibrium of composite strips along the x‐ and y‐axis. The displacement field so derived not only depends on the element co‐ordinates but is a function of extensional, bending–extensional coupling, bending and transverse shear stiffnesses as well. The element assumes bi‐cubic polynomial distribution with sixteen generalized undetermined coefficients for the transverse displacement. The element stiffness matrix and load vector are computed numerically by employing 3×3 Gauss–Legendre product rules. The element is found to be devoid of shear locking and does not exhibit any spurious modes. A series of numerical examples are solved to demonstrate the efficacy of the proposed element. Further, the element is found to yield consistently accurate results even with coarse mesh sizes over a wide range of thick plate regimes. Copyright © 2000 John Wiley & Sons, Ltd.     

p‐version of the generalized FEM using mesh‐based handbooks with applications to multiscale problems

In this paper, we analyse the p‐convergence of a new version of the generalized finite element method (generalized FEM or GFEM) which employs mesh‐based handbook functions which are solutions of boundary value problems in domains extracted from vertex patches of the employed mesh and are pasted into the global approximation by the partition of unity method (PUM). We show that the p‐version of our GFEM is capable of achieving very high accuracy for multiscale problems which may be impossible to solve using the standard FEM. We analyse the effect of the main factors affecting the accuracy of the method namely: (a) The data and the buffer included in the handbook domains, and (b) The accuracy of the numerical construction of the handbook functions. We illustrate the robustness of the method by employing as model problem the Laplacian in a domain with a large number of closely spaced voids. Similar robustness can be expected for problems of heat‐conduction and elasticity set in domains with a large number of closely spaced voids, cracks, inclusions, etc. Copyright © 2004 John Wiley & Sons, Ltd.     

Construction of shape functions for the h- and p-versions of the FEM using tensorial product

This paper presents an uniform and unified approach to construct h- and p-shape functions for quadrilaterals, triangles, hexahedral and tetrahedral based on the tensorial product of one-dimensional Lagrange and Jacobi polynomials. The approach uses indices to denote the one-dimensional polynomials in each tensorization direction. The appropriate manipulation of the indices allows to obtain hierarchical or non-hierarchical and inter-element C⁰ continuous or non-continuous bases. For the one-dimensional elements, quadrilaterals, triangles and hexahedral, the optimal weights of the Jacobi polynomials are determined, the sparsity profiles of the local mass and stiffness matrices plotted and the condition numbers calculated. A brief discussion of the use of sum factorization and computational implementation is considered. Copyright © 2006 John Wiley & Sons, Ltd.     

FFT‐based spectral element analysis for the linear continuum dynamic systems subjected to arbitrary initial conditions by using the pseudo‐force method

In this paper, a fast Fourier transform (FFT)‐based spectral element method (SEM) is developed for the linear continuum dynamic systems subjected to arbitrary, non‐null initial conditions. In the FFT‐based SEM, the original equations of motion subjected to arbitrary initial conditions are transformed into a new set of equations of motion subjected to completely null initial conditions by using the pseudo‐force method so that the conventional spectral element analysis can be applied to obtain desired dynamic responses. A simply supported beam and a cantilevered beam are considered as the illustrative problems to evaluate the FFT‐based SEM. The dynamic responses obtained by using the FFT‐based SEM are shown to be in good agreement with the analytical solutions obtained by using the mode superposition method. Copyright © 2007 John Wiley & Sons, Ltd.     

A meshfree‐enriched finite element method for compressible and near‐incompressible elasticity

In this paper, a two‐dimensional displacement‐based meshfree‐enriched FEM (ME‐FEM) is presented for the linear analysis of compressible and near‐incompressible planar elasticity. The ME‐FEM element is established by injecting a first‐order convex meshfree approximation into a low‐order finite element with an additional node. The convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the Kronecker‐delta property on the element boundaries. The gradient matrix of ME‐FEM element satisfies the integration constraint for nodal integration and the resultant ME‐FEM formulation is shown to pass the constant stress test for the compressible media. The ME‐FEM interpolation is an element‐wise meshfree interpolation and is proven to be discrete divergence‐free in the incompressible limit. To prevent possible pressure oscillation in the near‐incompressible problems, an area‐weighted strain smoothing scheme incorporated with the divergence‐free ME‐FEM interpolation is introduced to provide the smoothing on strains and pressure. With this smoothed strain field, the discrete equations are derived based on a modified Hu–Washizu variational principle. Several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near‐incompressible problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Two‐scale shear band evolution by local partition of unity

We introduce a methodology to model shear band evolution in the quasi‐static regime using the extended finite element method. We enrich the finite element polynomial displacement field with a fine scale function, which models the high displacement gradient in the shear band. For this purpose we use a local partition of unity and a parameterized displacement enrichment based on closed form solutions for one‐dimensional shear bands. A stabilized consistent penalty method is used to circumvent locking in the regularized elasto‐viscoplastic plane‐strain regime and to guarantee element stability. The loss of stability of the boundary value problem is used as an indicator of shear band initiation point and direction. Shear band development examples are shown, illustrating the capabilities of the method to track shear band evolution and strains as high as 1000%. Copyright © 2005 John Wiley & Sons, Ltd.     

A quasi‐static crack growth simulation based on the singular ES‐FEM

In the edge‐based smoothed finite element method (ES‐FEM), one needs only the assumed displacement values (not the derivatives) on the boundary of the edge‐based smoothing domains to compute the stiffness matrix of the system. Adopting this important feature, a five‐node crack‐tip element is employed in this paper to produce a proper stress singularity near the crack tip based on a basic mesh of linear triangular elements that can be generated automatically for problems with complicated geometries. The singular ES‐FEM is then formulated and used to simulate the crack propagation in various settings, using a largely coarse mesh with a few layers of fine mesh near the crack tip. The results demonstrate that the singular ES‐FEM is much more accurate than X‐FEM and the existing FEM. Moreover, the excellent agreement between numerical results and the reference observations shows that the singular ES‐FEM offers an efficient and high‐quality solution for crack propagation problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Arbitrary discontinuities in space–time finite elements by level sets and X‐FEM

An enriched finite element method with arbitrary discontinuities in space–time is presented. The discontinuities are treated by the extended finite element method (X‐FEM), which uses a local partition of unity enrichment to introduce discontinuities along a moving hyper‐surface which is described by level sets. A space–time weak form for conservation laws is developed where the Rankine–Hugoniot jump conditions are natural conditions of the weak form. The method is illustrated in the solution of first order hyperbolic equations and applied to linear first order wave and non‐linear Burgers' equations. By capturing the discontinuity in time as well as space, results are improved over capturing the discontinuity in space alone and the method is remarkably accurate. Implications to standard semi‐discretization X‐FEM formulations are also discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Application of the X‐FEM to the fracture of piezoelectric materials

This paper presents an application of the extended finite element method (X‐FEM) to the analysis of fracture in piezoelectric materials. These materials are increasingly used in actuators and sensors. New applications can be found as constituents of smart composites for adaptive electromechanical structures. Under in service loading, phenomena of crack initiation and propagation may occur due to high electromechanical field concentrations. In the past few years, the X‐FEM has been applied mostly to model cracks in structural materials. The present paper focuses at first on the definition of new enrichment functions suitable for cracks in piezoelectric structures. At second, generalized domain integrals are used for the determination of crack tip parameters. The approach is based on specific asymptotic crack tip solutions, derived for piezoelectric materials. We present convergence results in the energy norm and for the stress intensity factors, in various settings. Copyright © 2008 John Wiley & Sons, Ltd.     

集中荷载作用下弹性扭转约束层合浅拱的非线性面内稳定

该文以3D打印材料ABS-M30作为试验载体,开展了拱脚沉降下3D打印拱的非线性失稳研究。基于最小势能原理推导了失稳临界荷载的解析表达式,得到了拱脚竖向和水平变位下拱径向位移沿拱轴线的分布图;设计了可控制拱脚沉降的加载系统,试验得到了拱在加载过程中的平衡路径,并通过有限元模拟对解析与试验结果进行了验证;分析了拱脚沉降量、矢跨比和长细比对3D打印拱失稳临界荷载的影响。研究结果表明：非线性失稳临界荷载随着拱脚沉降量的增大而减小;在拱脚沉降量一定的前提下,非线性失稳荷载随着矢跨比的增大而增大,随着长细比的增大而减小,且长细比的影响最为显著。     

Extended rotation‐free plate and beam elements with shear deformation effects

This paper describes a methodology for extending rotation‐free plate and beam elements to accounting for transverse shear deformation effects. The ingredients for the element formulation are a Hu–Washizu‐type mixed functional, a linear interpolation for the deflection and the shear angles over standard finite elements and a finite volume approach for computing the bending moments and the curvatures over a patch of elements. As a first application of the general procedure, we present an extension of the three‐noded rotation‐free basic plate triangle (BPT) originally developed for thin plate analysis to account for shear deformation effects of relevance for thick plates and composite‐laminated plates. The nodal deflection degrees of freedom (DOFs) of the original BPT element are enhanced with the two shear deformation angles. This allows to compute the bending and shear deformation energies leading to a simple triangular plate element with three DOFs per node (termed BPT+ element). For the thin plate case, the shear angles vanish and the element reproduces the good behaviour of the original thin BPT element. As a consequence the element is applicable to thick and thin plate situations without exhibiting shear locking effects. The numerical solution for the thick case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin BPT element. A two‐noded rotation‐free beam element termed CCB+ applicable to slender and thick beams is derived as a particular case of the plate formulation. The examples presented show the robustness and accuracy of the BPT+ and the CCB+ elements for thick and thin plate and beam problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Axisymmetric modal analysis of liquid‐storage tanks considering compressibility effects

In this paper, we address analytical and numerical studies on the free vibration of fluid–structure interaction problems considering the fluid compressibility. According to the separation of variables together with the boundary condition enforcement, we first derive a compressible‐fluid velocity potential function. Next, we split the structure region into the wet and dry parts, for which we apply the Novozhilov thin shell theory. Combining two dynamic displacement fields, for two split structure parts, using the displacement compatibility conditions, we finally obtain a simultaneous equation system for computing natural frequencies and modes. According to the derived analytical formulae, we compute natural frequencies and modes, stress resultants, together with the comparison with the FEM analysis and the incompressible case. Numerical results show, compared to the incompressible case, that the compressible case produces lower natural frequencies and, furthermore, the relative difference is influenced by the slenderness of tanks and the relative liquid fill height. Copyright © 2002 John Wiley & Sons, Ltd.     

Surfactant‐Free Shape Control of Gold Nanoparticles Enabled by Unified Theoretical Framework of Nanocrystal Synthesis

Gold nanoparticles have unique properties that are highly dependent on their shape and size. Synthetic methods that enable precise control over nanoparticle morphology currently require shape‐directing agents such as surfactants or polymers that force growth in a particular direction by adsorbing to specific crystal facets. These auxiliary reagents passivate the nanoparticles' surface, and thus decrease their performance in applications like catalysis and surface‐enhanced Raman scattering. Here, a surfactant‐ and polymer‐free approach to achieving high‐performance gold nanoparticles is reported. A theoretical framework to elucidate the growth mechanism of nanoparticles in surfactant‐free media is developed and it is applied to identify strategies for shape‐controlled syntheses. Using the results of the analyses, a simple, green‐chemistry synthesis of the four most commonly used morphologies: nanostars, nanospheres, nanorods, and nanoplates is designed. The nanoparticles synthesized by this method outperform analogous particles with surfactant and polymer coatings in both catalysis and surface‐enhanced Raman scattering.     

Non‐linear exact geometry 12‐node solid‐shell element with three translational degrees of freedom per node

This paper presents the finite rotation exact geometry (EG) 12‐node solid‐shell element with 36 displacement degrees of freedom. The term ‘EG’ reflects the fact that coefficients of the first and second fundamental forms of the reference surface and Christoffel symbols are taken exactly at each element node. The finite element formulation developed is based on the 9‐parameter shell model by employing a new concept of sampling surfaces (S‐surfaces) inside the shell body. We introduce three S‐surfaces, namely, bottom, middle and top, and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows one to represent the finite rotation higher order EG solid‐shell element formulation in a very compact form and to derive the strain–displacement relationships, which are objective, that is, invariant under arbitrarily large rigid‐body shell motions in convected curvilinear coordinates. The tangent stiffness matrix is evaluated by using 3D analytical integration and the explicit presentation of this matrix is given. The latter is unusual for the non‐linear EG shell element formulation. Copyright © 2011 John Wiley & Sons, Ltd.     

High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

Based on the differential quadrature (DQ) rule, the Gauss Lobatto quadrature rule and the variational principle, a DQ finite element method (DQFEM) is proposed for the free vibration analysis of thin plates. The DQFEM is a highly accurate and rapidly converging approach, and is distinct from the differential quadrature element method (DQEM) and the quadrature element method (QEM) by employing the function values themselves in the trial function for the title problem. The DQFEM, without using shape functions, essentially combines the high accuracy of the differential quadrature method (DQM) with the generality of the standard finite element formulation, and has superior accuracy to the standard FEM and FDM, and superior efficiency to the p‐version FEM and QEM in calculating the stiffness and mass matrices. By incorporating the reformulated DQ rules for general curvilinear quadrilaterals domains into the DQFEM, a curvilinear quadrilateral DQ finite plate element is also proposed. The inter‐element compatibility conditions as well as multiple boundary conditions can be implemented, simply and conveniently as in FEM, through modifying the nodal parameters when required at boundary grid points using the DQ rules. Thus, the DQFEM is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter‐element compatibilities. A series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the DQFEM. Copyright © 2009 John Wiley & Sons, Ltd.