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.     

A nonlinear plate finite element formulation for shape memory alloy applications

The aim of the present work is to develop a new finite element model for the finite strain analysis of plate structures constituted of shape memory alloy (SMA) material. A three‐dimensional constitutive model for shape memory alloys able to reproduce the special thermomechanical behavior of SMA characterized by pseudoelasticity and shape memory effects is adopted. The finite strain constitutive model is thermodynamically consistent and is completely formulated in the reference configuration. A two‐dimensional plate theory is proposed based on a tensor element shape function formulation. The displacement field is expressed in terms of increasing powers of the transverse coordinate. The equilibrium statement is formulated on the basis of the virtual displacement principle in a total Lagrangian format. The proposed displacement formulation is particularly suitable for the simple derivation of high‐order finite elements. Numerical applications are performed to assess the efficiency and locking performance of the proposed plate finite element. Some additional numerical examples are carried out to study the accuracy and robustness of the proposed computational technique and its capability of describing the structural response of SMA devices. Copyright © 2011 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.     

Refined triangular discrete Kirchhoff plate element for thin plate bending,vibration and buckling analysis

A refined triangular discrete Kirchhoff thin plate bending element RDKT which can be used to improve the original triangular discrete Kirchhoff thin plate bending element DKT is presented. In order to improve the accuracy of the analysis a simple explicit expression of a refined constant strain matrix with an adjustable constant can be introduced into its formulation. The new element displacement function can be used to formulate a mass matrix called combined mass matrix for calculation of the natural frequency and in the same way a combined geometric stiffness matrix can be obtained for buckling analysis. Numerical examples are presented to show that the present methods indeed, can improve the accuracy of thin plate bending, vibration and buckling analysis. © 1998 John Wiley & Sons, Ltd.     

Fully tensorial nodal and modal shape functions for triangles and tetrahedra

This paper presents nodal and modal shape functions for triangle and tetrahedron finite elements. The functions are constructed based on the fully tensorial expansions of one‐dimensional polynomials expressed in barycentric co‐ordinates. The nodal functions obtained from the application of the tensorial procedure are the standard h‐Lagrange shape functions presented in the literature. The modal shape functions use Jacobi polynomials and have a natural global C⁰ inter‐element continuity. An efficient Gauss–Jacobi numerical integration procedure is also presented to decrease the number of points for the consistent integration of the element matrices. An example illustrates the approximation properties of the modal functions. Copyright © 2005 John Wiley & Sons, Ltd.     

高速旋转柔性矩形薄板的动力学建模和近似算法

研究了做高速旋转柔性矩形薄板的耦合动力学建模理论和模态截断法的应用.从连续介质力学中关于柔性薄板的变形理论出发,找出了由于在结构动力学中对无大范围运动柔性薄板的动力学性质影响很小而被忽略的变形量.基于Jourdain速度变分原理,先推导出做高速旋转薄板的动力学连续变分方程,再用有限元法对柔性薄板进行离散.因为用有限元离散时,柔性薄板的广义坐标规模较大,故仿真计算需要时间较长,所以模态截断法被用来缩减广义坐标数量,提高计算效率.此外通过对有限元法和模态截断法的计算结果进行比较,揭示了当矩形薄板作高速旋转时,模态截断法截取低阶模态时会引起误差,选取更高的模态可以用来提高计算精度,通过数值对比得到了模态截断的规律.     

An iterative regularization method for inverse phonon radiative transport problem in nanoscale thin films

An inverse phonon radiative transport problem with an alternative form of adjoint equation is solved in this study by using conjugate gradient method (CGM) to estimate the unknown boundary temperature distributions, based on the phonon intensity measurements. The CGM in dealing with the present integro‐differential governing equations is not as straightforward as for the normal differential equations; special treatments are needed to overcome the difficulties. Results obtained in this inverse analysis will be justified based on the numerical experiments where two different unknown temperature (or phonon intensity) distributions are to be determined. Finally, it is shown that accurate boundary temperatures can always be obtained with CGM. Copyright © 2006 John Wiley & Sons, Ltd.     

Embedded kinematic boundary conditions for thin plate bending by Nitsche's approach

A stabilized variational formulation, based on Nitsche's method for enforcing boundary constraints, leads to an efficient procedure for embedding kinematic boundary conditions in thin plate bending. The absence of kinematic admissibility constraints allows the use of non‐conforming meshes with non‐interpolatory approximations, thereby providing added flexibility in addressing the C¹‐continuity requirements typical of these problems. Work‐conjugate pairs weakly enforce kinematic boundary conditions. The pointwise enforcement of corner deflections is key to good performance in the presence of corners. Stabilization parameters are determined from local generalized eigenvalue problems, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic C² B‐splines, exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameters. Copyright © 2012 John Wiley & Sons, Ltd.     

An analytical model for the vibration of a composite plate containing an embedded periodic shape memory alloy structure

This paper explores the integration of a periodic repeating arrangement of shape memory alloy (SMAs) within a composite plate, with a view to active control of the vibrations of the plate by means of a controllable activation strategy for the SMA elements. The benefits of this configuration are that ‘antagonistic’ operation of SMAs on the plate allows the significantly longer cooling time constant of previously activated elements to be shortened by means of active elements working against them during that phase. This concept dramatically shortens the cooling time constant and brings it into the same order of magnitude of the heating phase. The paper examines the mathematical modelling of such a plate, and offers an approximate analytical solution by means of a hybrid WKB–Galerkin method. The antagonistic operation of the system is represented mathematically by terms in which the stiffness and damping are both time dependent. Therefore the equation of motion contains terms with time variant coefficients and is impossible to solve without recourse to specialised methods. Comparisons with numerical methods are given and it is shown that good similarity can be obtained for judicious choice of practical values for the time variant stiffness and damping functions.     

Photoelastic inverse problem solution for a biaxially loaded infinite plate with a hole

This paper presents an inverse problem solution for the case of photoelastic isochromatic fringes around the hole of a biaxially uniformly loaded infinite plate for two problems. Problem 1 relates to an infinite plate in which the circular hole is drilled first and then the loads are applied. Problem 2 is the residual stress problem in which the hole is drilled after the biaxial load is applied to the infinite plate. This "hole method" solution may be used for all cases of biaxial, far field uniform loading conditions.     

Improvement of numerical modeling in the solution of static and transient dynamic problems using finite element method based on spherical Hankel shape functions

In this paper, finite element method is reformulated using new shape functions to approximate the state variables (ie, displacement field and its derivatives) and inhomogeneous term (ie, inertia term) of Navier's differential equation. These shape functions and corresponding elements are called spherical Hankel hereafter. It is possible for these elements to satisfy the polynomial and the first and second kind of Bessel function fields simultaneously, while the classic Lagrange elements can only satisfy polynomial ones. These shape functions are so robust that with least degrees of freedom, much better results can be achieved in comparison with classic Lagrange ones. It is interesting that no Runge phenomenon exists in the interpolation of proposed shape functions when going to higher degrees of freedom, while it may occur in classic Lagrange ones. Moreover, the spherical Hankel shape functions have a significant robustness in the approximation of folded surfaces. Five numerical examples related to the usage of suggested shape functions in finite element method in solving problems are studied, and their results are compared with those obtained from classic Lagrange shape functions and analytical solutions (if available) to show the efficiency and accuracy of the present method.     

Direct collocation method for identifying the initial conditions in the inverse wave problem using radial basis functions

A direct collocation method associated with explicit time integration using radial basis functions is proposed for identifying the initial conditions in the inverse problem of wave propagation. Optimum weights for the boundary conditions and additional condition are derived based on Lagrange’s multiplier method to achieve the prime convergence. Tikhonov regularization is introduced to improve the stability for the ill-posed system resulting from the noise, and the L-curve criterion is employed to select the optimum regularization parameter. No iteration scheme is required during the direct collocation computation which promotes the accuracy and stability for the solutions, while Galerkin-based methods demand the iteration procedure to deal with the inverse problems. High accuracy and good stability of the solution at very high noise level make this method a superior scheme for solving inverse problems.     

Piezoelectric study for a coated hole of quasi-polygonal shape in an infinite plate

This paper studies the piezoelectric problems for a coated hole of quasi-polygonal shape embedded in an infinite matrix subjected to electromechanical loadings. The electromechanical loadings considered in this work include a screw dislocation, a point force, a point charge, a far-field anti-plane shear and an in-plane electric field. Each component is assumed to be transversely isotropic medium belonging to a hexagonal crystal class 6 mm and poled in the x₃ direction. Based on the complex variable and analytical continuation method, the general expressions for the complex potentials can be derived in each medium. Numerical results are provided to show the effect of hole shape, the material combinations and the loading condition on the electro-elastic fields due to the presence of the coated film. The image force exerted on a dislocation, which can be used to probe the mobility of the screw dislocation, will be calculated by means of the generalized Peach-Koehler formula.     

New numerical approach in the analysis of a thin wire radiating over a lossy half-space

The finite element formulation for a half-space electric field integral equation is described. Sommerfeld integrals, appearing in the kernel of the integral equation are calculated by means of exponential approximations. This approach shows advantages over the usual techniques. Obtained results are compared with other results available.     

无风环境下振动薄平板断面的非线性气动效应研究

基于随机拉格朗日-欧拉(ALE)法和弱耦合方法,建立了断面风振响应计算的数值模型。提出了无风环境下振动断面的气动力数学表达式,模拟了薄平板断面在无风环境下的自由振动响应,进而计算了断面的非线性振动频率和阻尼比。结果表明:提出的气动力模型能够有效的描述无风环境下作用在振动断面上的非定常气动力;无风环境下振动中的主梁断面,竖向振动比扭转振动对周围空气的干扰作用更大;若忽略无风环境下的气动效应,将会带来一定的误差;初始激励对无风环境下主梁断面的气动效应有较大的影响。     

A new approach for displacement functions of a curved Timoshenko beam element in motions normal to its initial plane

In existing literature, either analytical methods or numerical methods, the formulations for free vibration analysis of circularly curved beams normal to its initial plane are somewhat complicated, particularly if the effects of both shear deformation (SD) and rotary inertia (RI) are considered. It is hoped that the simple approach presented in this paper may improve the above‐mentioned drawback of the existing techniques. First, the three functions for axial (or normal to plane) displacement and rotational angles about radial and circumferential (or tangential) axes of a curved beam element were assumed. Since each function consists of six integration constants, one has 18 unknown constants for the three assumed displacement functions. Next, from the last three displacement functions, the three force–displacement differential equations and the three static equilibrium equations for the arc element, one obtained three polynomial expressions. Equating to zero the coefficients of the terms in each of the last three expressions, respectively, one obtained 17 simultaneous equations as functions of the 18 unknown constants. Excluding the five dependent ones among the last 17 equations, one obtained 12 independent simultaneous equations. Solving the last 12 independent equations, one obtained a unique solution in terms of six unknown constants. Finally, imposing the six boundary conditions at the two ends of an arc element, one determined the last six unknown constants and completely defined the three displacement functions. By means of the last displacement functions, one may calculate the shape functions, stiffness matrix, mass matrix and external loading vector for each arc element and then perform the free and forced vibration analyses of the entire curved beam. Good agreement between the results of this paper and those of the existing literature confirms the reliability of the presented theory. Copyright © 2005 John Wiley & Sons, Ltd.     

Global 2D digital image correlation for motion estimation in a finite element framework: a variational formulation and a regularized,pyramidal, multi‐grid implementation

In this study, the inverse problem of reconstructing the in‐plane (2D) displacements of a monitored surface through a sequence of two‐dimensional digital images, severely ill‐posed in Hadamard's sense, is deeply investigated. A novel variational formulation is presented for the continuum 2D digital image correlation problem, and critical issues such as semi‐coercivity and solution multiplicity are discussed by functional analysis tools. In the framework of a Galerkin, finite element discretization of the displacement field, a robust implementation for 2D digital image correlation is outlined, aiming to attenuate the spurious oscillations which corrupt the deformation scenario, especially when very fine meshes are adopted. Recourse is made to a hierarchical family of grids linked by suitable restriction and prolongation operators and defined over an image pyramid. Multi‐grid cycles are performed ascending and descending along the pyramid, with only one Newton iteration per level irrespective of the tolerance satisfaction, as if the problem were linear. At each level, the conventional least‐square matching functional is herein enriched by a Tychonoff regularization provision, preserving the solution against an unstable response. The algorithm is assessed on the basis of both synthetic and truly experimental image pairs. Copyright © 2013 John Wiley & Sons, Ltd.     

A generalized Ritz method for partial differential equations in domains of arbitrary geometry using global shape functions

A boundary element method (BEM)-based variational method is presented for the solution of elliptic PDEs describing the mechanical response of general inhomogeneous anisotropic bodies of arbitrary geometry. The equations, which in general have variable coefficients, may be linear or nonlinear. Using the concept of the analog equation of Katsikadelis the original equation is converted into a linear membrane (Poisson) or a linear plate (biharmonic) equation, depending on the order of the PDE under a fictitious load, which is approximated with radial basis function series of multiquadric (MQ) type. The integral representation of the solution of the substitute equation yields shape functions, which are global and satisfy both essential and natural boundary conditions, hence the name generalized Ritz method. The minimization of the functional that produces the PDE as the associated Euler–Lagrange equation yields not only the Ritz coefficients but also permits the evaluation of optimal values for the shape parameters of the MQs as well as optimal position of their centers, minimizing thus the error. If a functional does not exists or cannot be constructed as it is the usual case of nonlinear PDEs, the Galerkin principle can be applied. Since the arising domain integrals are converted into boundary line integrals, the method is boundary-only and, therefore, it maintains all the advantages of the pure BEM. Example problems are studied, which illustrate the method and demonstrate its efficiency and great accuracy.     

Coherent state path integral approach to multi-time correlation functions and photon emission in a semiconductor microcavity

In this paper by using the coherent state path integral field theory approach, we calculate the grand canonical partition function of an interacting combined system in the presence of the relevant source terms. It allows us to calculate multi-time correlation functions of interacting systems without using the quantum regression theorem. Then, we investigate the power spectrum and the second-order correlation function of the emitted photons from a microcavity in the presence of excitations of a semiconductor quantum well. By using the Hubbard–Stratonovich transformation, we investigate the effects of reservoir, detuning, the Coulomb interaction and the phase space filling on the power spectrum and the second-order correlation function of the emitted photons.     

Numerical algorithms for cyclic phase transformation hysteresis in a shape memory plate subject to axisymmetric plane stress

We present numerical algorithms for calculating stress fields in an annulus composed of a shape memory material under conditions of quasi‐static edge loading at constant temperature. The algorithms track the material microstructure in terms of the volume fraction of austenite (A) and martensite (M), the latter of which provides a transformation strain. The dependence on load path imparts significant hysteresis in the stress induced transformation between A and M. A previous study that was restricted to proportional loading in the direction of forward transformation (J. Appl. Mech. 2005; 72 :44–53) is here generalized to consider arbitrary loadings. The shooting algorithm that was robust for the previously considered proportional loadings is found to be subject to numerical instability for the most general transformation possibilities considered here. This motivates the development of an alternative iterated mapping algorithm that is found to generate a robust semi‐analytical finite difference procedure. The algorithm efficiently determines the operative transformation type, as is illustrated in cases where forward and reverse loading are occurring simultaneously at different plate locations. At those locations where phase transformation is inactive, the algorithm continues to account for martensite reorientation that alters the local transformation strain. Copyright © 2006 John Wiley & Sons, Ltd.