Post‐buckling analysis of imperfect laminates using finite strips based on a higher‐order plate theory

A finite strip procedure has been developed for the post‐buckling analysis of composite laminates when subjected to progressive end shortening. The finite strips are developed based on a higher‐order shear deformation plate theory and there are nine variables at each nodal line. Initial imperfection expressed in the form of suitable trigonometric function is allowed. Examples including isotropic plates and laminates with arbitrary lay‐up arrangement are presented. Numerical results for laminates with and without initial imperfection are used to illustrate the effect of imperfection. Copyright © 2003 John Wiley & Sons, Ltd.     

Mixed plate bending elements based on least‐squares formulation

A finite element formulation for the bending of thin and thick plates based on least‐squares variational principles is presented. Finite element models for both the classical plate theory and the first‐order shear deformation plate theory (also known as the Kirchhoff and Mindlin plate theories, respectively) are considered. High‐order nodal expansions are used to construct the discrete finite element model based on the least‐squares formulation. Exponentially fast decay of the least‐squares functional, which is constructed using the L₂ norms of the equations residuals, is verified for increasing order of the nodal expansions. Numerical examples for the bending of circular, rectangular and skew plates with various boundary conditions and plate thickness are presented to demonstrate the predictive capability and robustness of the new plate bending elements. Plate bending elements based on this formulation are shown to be insensitive to both shear‐locking and geometric distortions. Copyright © 2004 John Wiley & Sons, Ltd.     

Distortion‐immune nine‐node displacement‐based quadrilateral thick plate finite elements that satisfy constant‐bending patch test

We employ the linked interpolation concept to develop two higher‐order nine‐node quadrilateral plate finite elements with curved sides that pass the constant bending patch test for arbitrary node positions. The linked interpolation for the plate displacements is expanded with three bubble parameters to get polynomial completeness necessary to satisfy the patch test. In contrast to some other techniques, the elements developed in this way retain a symmetric stiffness matrix at a marginal computational expense at the element level. The new elements generated using this concept are tested on several examples with curved sides or some other kind of geometric distortion. Copyright © 2014 John Wiley & Sons, Ltd.     

Topology optimization of thin‐walled box beam structures based on the higher‐order beam theory

The investigation aims to formulate ground‐structure based topology optimization approach by using a higher‐order beam theory suitable for thin‐walled box beam structures. While earlier studies use the Timoshenko or Euler beams to form a ground‐structure, they are not suitable for a structure consisting of thin‐walled closed beams. The higher‐order beam theory takes into an additional account sectional deformations of a thin‐walled box beam such as warping and distortion. Therefore, a method to connect ground beams at a joint and a technique to represent different joint connectivity states should be investigated for streamlined topology optimization. Several numerical case studies involving different loading and boundary conditions are considered to show the effectiveness of employing a higher‐order beam theory for the ground‐structure based topology optimization of thin‐walled box beam structures. Through the numerical results, this work shows significant difference between optimized beam layouts based on the Timoshenko beam theory and those based on a more accurate higher‐order beam theory for a structure consisting of thin‐walled box beams. Copyright © 2015 John Wiley & Sons, Ltd.     

An adaptive finite element method for second‐order plate theory

We present a discontinuous finite element method for the Kirchhoff plate model with membrane stresses. The method is based on P₂‐approximations on simplices for the out‐of‐plane deformations, using C⁰‐continuous approximations. We derive a posteriori error estimates for linear functionals of the error and give some numerical examples. Copyright © 2009 John Wiley & Sons, Ltd.     

An interpolation‐based fast multipole method for higher‐order boundary elements on parametric surfaces

In this article, a black‐box higher‐order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed. Such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis. Due to the exact surface representation, the rate of convergence of higher‐order methods is not limited by approximation errors of the surface. An element‐wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying Galerkin method. By performing the interpolation for the fast multipole method directly on the reference domain, the cost complexity in the polynomial degree is reduced by one order. This gain is independent of the application of either ‐ or ‐matrices. In fact, several simplifications in the construction of ‐matrices are pointed out, which are a by‐product of the surface representation. Extensive numerical examples are provided in order to quantify and qualify the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Refined global–local higher‐order theory and finite element for laminated plates

Based on completely three‐dimensional elasticity theory, a refined global–local higher‐order theory is presented as enhanced version of the classical global–local theory proposed by Li and Liu (Int. J. Numer. Meth. Engng. 1997; 40 :1197–1212), in which the effect of transverse normal deformation is enhanced. Compared with the previous higher‐order theory, the refined theory offers some valuable improvements these are able to predict accurately response of laminated plates subjected to thermal loading of uniform temperature. However, the previous higher‐order theory will encounter difficulty for this problem. A refined three‐noded triangular element satisfied the requirement of C¹ weak‐continuity conditions in the inter‐element is also presented. The results of numerical examples of moderately thick laminated plates and even thick plates with span/thickness ratios L/h = 2 are given to show that in‐plane stresses and transverse shear stresses can be reasonably predicted by the direct constitutive equation approach without smooth technique. In order to accurately obtain transverse normal stresses, the local equilibrium equation approach in one element is employed herein. Copyright © 2006 John Wiley & Sons, Ltd.     

A unified 3D‐based technique for plate bending analysis using scaled boundary finite element method

This paper presents a unified technique for solving the plate bending problems by extending the scaled boundary finite element method. The formulation is based on the three‐dimensional governing equation without enforcing the kinematics of plate theory. Only the in‐plane dimensions are discretised into finite elements. Any two‐dimensional displacement‐based elements can be employed. The solution along the thickness is expressed analytically by using a matrix function. The proposed technique is consistent with the three‐dimensional theory and applicable to both thick and thin plates without exhibiting the numerical locking phenomenon. Moreover, the use of higher order spectral elements allows the proposed technique to better represent curved boundaries and to achieve high accuracy and fast convergence. Numerical examples of various plate structures with different thickness‐to‐length ratios demonstrate the applicability and accuracy of the proposed technique. Copyright © 2012 John Wiley & Sons, Ltd.     

EAS concept for higher‐order finite shell elements to eliminate volumetric locking

The deficiency of volumetric locking phenomena in finite elements using higher‐order shell element formulations based on Lagrangean polynomials and a linear finite shell kinematics cannot be avoided by the existent enhanced assumed strain (EAS) concept established for low‐order elements. In this paper a consistent modification of the EAS concept is proposed to extend its applicability to higher‐order shell elements. This modification, affecting the transversal normal strain for polynomial orders p>1, eliminates pathological modes caused by volumetric locking. The efficiency of the proposed extended EAS method is demonstrated by means of eigenvalue analyses and two representative numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.     

Free vibration analysis of bimodulus laminated plates using higher‐order plate element

Abstract

A C° isoparametric higher‐order plate element is developed to analyze the free vibration of bimodulus laminated plates. The equations of motion for the higher‐order plate theory are also derived variationally. The natural frequencies and neutral surface locations are determined for benchmark problems. The numerical results are compared to available analytical solutions, and excellent agreement is observed. Obviously, the present formulation is more accurate than the first‐order theory.     

Conformal higher‐order remeshing schemes for implicitly defined interface problems

A new higher‐order accurate method is proposed that combines the advantages of the classical p‐version of the FEM on body‐fitted meshes with embedded domain methods. A background mesh composed by higher‐order Lagrange elements is used. Boundaries and interfaces are described implicitly by the level set method and are within elements. In the elements cut by the boundaries or interfaces, an automatic decomposition into higher‐order accurate sub‐elements is realized. Therefore, the zero level sets are detected and meshed in a first step, which is called reconstruction. Then, based on the topological situation in the cut element, higher‐order sub‐elements are mapped to the two sides of the boundary or interface. The quality of the reconstruction and the mapping largely determines the properties of the resulting, automatically generated conforming mesh. It is found that optimal convergence rates are possible although the resulting sub‐elements are not always well‐shaped. Copyright © 2016 John Wiley & Sons, Ltd.     

A cell‐based smoothed discrete shear gap method (CS‐FEM‐DSG3) based on the C0‐type higher‐order shear deformation theory for dynamic responses of Mindlin plates on viscoelastic foundations subjected to a moving sprung vehicle

A cell‐based smoothed discrete shear gap method (CS‐FEM‐DSG3) based on the first‐order shear deformation theory (FSDT) was recently proposed for static and dynamic analyses of Mindlin plates. In this paper, the CS‐FEM‐DSG3 is extended to the C⁰‐type higher‐order shear deformation plate theory (C⁰‐HSDT) and is incorporated with damping–spring systems for dynamic responses of Mindlin plates on viscoelastic foundations subjected to a moving sprung vehicle. At each time step of dynamic analysis, one four‐step procedure is performed including the following: (1) transformation of the weight of a four‐wheel vehicle into the sprung masses at wheels; (2) dynamic analysis of the sprung mass of wheels to determine the contact forces; (3) transformation of the contact forces into loads at nodes of plate elements; and (4) dynamic analysis of the plate elements on viscoelastic foundations. The accuracy and reliability of the proposed method are verified by comparing its numerical solutions with those of other available numerical results. Copyright © 2014 John Wiley & Sons, Ltd.     

基于三阶剪切变形理论的压电功能梯度板静力学等几何分析

针对压电功能梯度板的静力学问题,建立了一种基于三阶剪切变形理论的等几何分析求解方法.其中,定义功能梯度板的材料属性为板厚方向的幂函数分布,并假设压电功能梯度板中的机械位移场与电势场相互独立.利用压电材料的第二类本构方程以及哈密顿变分原理,推导出压电功能梯度板的相关等几何有限元方程.在压电功能梯度板的自由振动分析中,研究...     

A new triangular element to model inter‐laminar shear stress continuous plate theory

An efficient triangular element based on an inter‐laminar shear stress continuous plate theory is developed and applied to the analysis of composite and sandwich plates under different situations to study the performance of the element. The plate theory represents parabolic through thickness variation of transverse shear stresses where the continuity condition of these stresses are satisfied at the layer interfaces. It also satisfies transverse shear stress free condition at the top and bottom surfaces of the plate. The most attractive feature of the plate theory is that the basic unknowns are same as those used in first‐order shear deformation theory. The only problem lies with this elegant plate theory is found in its finite element implementation, as it requires C¹ continuity of transverse displacement at the element interfaces. This is a well‐known problem of thin plate elements, which is also found in some other refined plate theories. Although there are some elements based on these refined plate theories but the number of such elements is very few and they possess certain drawbacks in general. Keeping these aspects in view, an attempt has been made in this study to develop a six‐noded triangular element having equal degrees of freedom at each node. Copyright © 2004 John Wiley & Sons, Ltd.     

An improved discrete Kirchhoff quadrilateral element based on third‐order zigzag theory for static analysis of composite and sandwich plates

A new improved discrete Kirchhoff quadrilateral element based on the third‐order zigzag theory is developed for the static analysis of composite and sandwich plates. The element has seven degrees of freedom per node, namely, the three displacements, two rotations and two transverse shear strain components at the mid‐surface. The usual requirement of C¹ continuity of interpolation functions of the deflection in the third‐order zigzag theory is circumvented by employing the improved discrete Kirchhoff constraint technique. The element is free from the shear locking. The finite element formulation and the computer program are validated by comparing the results for simply supported plate with the analytical Navier solution of the zigzag theory. Comparison of the present results with those using other available elements based on zigzag theories for composite and sandwich plates establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The accuracy of the zigzag theory is assessed by comparing the finite element results of the square all‐round clamped composite plates with the converged three‐dimensional finite element solution obtained using ABAQUS. The comparisons also establish the superiority of the zigzag theory over the smeared third‐order theory having the same number of degrees of freedom. Copyright © 2006 John Wiley & Sons, Ltd.     

The tetrahedral finite cell method: Higher‐order immersogeometric analysis on adaptive non‐boundary‐fitted meshes

The finite cell method (FCM) is an immersed domain finite element method that combines higher‐order non‐boundary‐fitted meshes, weak enforcement of Dirichlet boundary conditions, and adaptive quadrature based on recursive subdivision. Because of its ability to improve the geometric resolution of intersected elements, it can be characterized as an immersogeometric method. In this paper, we extend the FCM, so far only used with Cartesian hexahedral elements, to higher‐order non‐boundary‐fitted tetrahedral meshes, based on a reformulation of the octree‐based subdivision algorithm for tetrahedral elements. We show that the resulting TetFCM scheme is fully accurate in an immersogeometric sense, that is, the solution fields achieve optimal and exponential rates of convergence for h‐refinement and p‐refinement, if the immersed geometry is resolved with sufficient accuracy. TetFCM can leverage the natural ability of tetrahedral elements for local mesh refinement in three dimensions. Its suitability for problems with sharp gradients and highly localized features is illustrated by the immersogeometric phase‐field fracture analysis of a human femur bone. Copyright © 2016 John Wiley & Sons, Ltd.     

Hygrothermal effects on multilayered composite plates using a refined higher order theory

A four-node quadrilateral plate element based on the global–local higher order theory (GLHOT) is proposed to study the response of laminated composite plates due to a variation in temperature and moisture concentrations. C⁰$C^0$ and C¹$C^1$ continuities are required in the transverse displacement functions as the first and the second derivatives are involved in the computation of the strain components in GLHOT [Int J Mech Sci, 2007; 49:1276–1288]. In this paper, a displacement function satisfying C⁰$C^0$ continuity is constructed by using the refined element method, and the discrete Kirchhoff quadrilateral thin plate element DKQ is employed for satisfying the requirement of C¹$C^1$ continuity. The effects of temperature and moisture concentrations on the material properties and the hygrothermal response of multilayered plates are studied, in contrast to most of the previous investigations in which the material properties are assumed to be independent of temperature. Hygrothermal response due to a variation in temperature and moisture concentrations has been studied for different material types sensitive to changing hygrothermal environment conditions. Numerical results suggest that temperature-dependent material properties ought to be used in the analysis of laminated plates subjected to hygrothermal loads.     

Analysis of high‐order finite elements for convected wave propagation

In this paper, we examine the performance of high‐order finite element methods (FEM) for aeroacoustic propagation, based on the convected Helmholtz equation. A methodology is presented to measure the dispersion and amplitude errors of the p‐FEM, including non‐interpolating shape functions, such as ‘bubble’ shape functions. A series of simple test cases are also presented to support the results of the dispersion analysis. The main conclusion is that the properties of p‐FEM that make its strength for standard acoustics (e.g., exponential p‐convergence, low dispersion error) remain present for flow acoustics as well. However, the flow has a noticeable effect on the accuracy of the numerical solution, even when the change in wavelength due to the mean flow is accounted for, and an approximation of the dispersion error is proposed to describe the influence of the mean flow. Also discussed is the so‐called aliasing effect, which can reduce the accuracy of the solution in the case of downstream propagation. This can be avoided by an appropriate choice of mesh resolution. Copyright © 2013 John Wiley & Sons, Ltd.