A critical analysis of quadratic beam finite elements

A new methodology of evaluation of C⁰ beam elements is presented. It is shown that, knowing the stiffness matrix of an arbitrary type of element, it is possible to create equivalent equilibrium conditions expressed in the form of one difference equation for a regular beam discretized by these elements. The study of the convergence of one difference equation gives an interpretation of the source of troubles occurring in low-order bending elements which is more convincing than the usually applied consideration of the conditioning of element stiffness matrices. A careful examination of quadratic Mindlin elements provides a very clear explanation of the shear locking essence in the Timoshenko beam. The presented method enables one to identify errors that appear also in the reduced integrated or constrained elements. For each type of analysed quadratic element an adequate difference equation is derived and compared with the exact one. Based on this comparison a simple method of corrections is proposed that completely eliminates the errors associated with the application of C⁰ bending elements.     

Mixed-interpolated finite elements for functionally graded cylindrical shells

In this paper cylindrical shells made of functionally graded materials (FGMs) are studied. A two-constituent material distribution through the thickness is considered, varying with a simple power rule of mixture. The equations governing the FGMs shells are determined using a variational formulation arising from the Naghdi theory. Moreover a strategy to achieve an improved transverse shear factor is investigated by energy equivalence. To approximate the problem a family of mixed-interpolated finite elements is used. It is based on a suitable reduction of the shear and membrane energy. Several numerical simulations are carried out in order to show the capability of the proposed elements to capture the properties of shells of various gradings, subjected to thermo-mechanical loads.     

Reduced modified quadratures for quadratic membrane finite elements

Reduced integration is frequently used in evaluating the element stiffness matrix of quadratically interpolated finite elements. Typical examples are the serendipity (Q8) and Lagrangian (Q9) membrane finite elements, for which a reduced 2 × 2 Gauss–Legendre integration rule is frequently used, as opposed to full 3 × 3 Gauss–Legendre integration. This ‘softens’ these element, thereby increasing accuracy, albeit at the introduction of spurious zero energy modes on the element level. This is in general not considered problematic for the ‘hourglass’ mode common to Q8 and Q9 elements, since this spurious mode is non‐communicable. The remaining two zero energy modes occurring in the Q9 element are indeed communicable. However, in topology optimization for instance, conditions may arise where the non‐communicable spurious mode associated with the elements becomes activated. To effectively suppress these modes altogether in elements employing quadratic interpolation fields, two modified quadratures are employed herein. For the Q8 and Q9 membrane elements, the respective rules are a five and an eight point rule. As compared to fully integrated elements, the new rules enhance element accuracy due to the introduction of soft, higher‐order deformation modes. A number of standard test problems reveal that element accuracy remains comparable to that of the under‐integrated counterparts. Copyright © 2004 John Wiley & Sons, Ltd.     

Elastostatic analysis of infinite solids using finite elements

A novel technique is developed to simulate the effects of an infinite elastic solid by using multiple springs having spatially varying stiffnesses. The spring constants are computed by numerical integration of classical solutions for point or line loads in an infinite or semi-infinite elastic mass. Under certain conditions, even the 'exact' values of spring constants may become negative at some nodes. A simple and highly effective algorithm is proposed to remove this computational difficulty. The technique is applied to the computation of displacements and stresses around underground openings. For a circular opening subjected to different stress conditions, spring constants computed by the proposed numerical integration technique are found to be 'identical' to their 'exact' values. Results obtained by the proposed technique for displacements and stresses around circular and non-circular openings are found to be in an excellent agreement with classical and boundary element solutions. The principal advantages of the proposed technique are that an unbounded solid may be simulated by a relatively very small finite model and a standard finite element code requires no modification for its implementation.     

Consistent multiscale analysis of heterogeneous thin plates with smoothed quadratic Hermite triangular elements

A consistent multiscale formulation is presented for the bending analysis of heterogeneous thin plate structures containing three dimensional reinforcements with in-plane periodicity. A multiscale asymptotic expansion of the displacement field is proposed to represent the in-plane periodicity, in which the microscopic and macroscopic thickness coordinates are set to be identical. This multiscale displacement expansion yields a local three dimensional unit cell problem and a global homogenized thin plate problem. The local unit cell problem is discretized with the tri-linear hexahedral elements to extract the homogenized material properties. The characteristic macroscopic deformation modes corresponding to the in-plane membrane deformations and out of plane bending deformations are discussed in detail. Thereafter the homogenized material properties are employed for the analysis of global homogenized thin plate with a smoothed quadratic Hermite triangular element formulation. The quadratic Hermite triangular element provides a complete C¹ approximation that is very desirable for thin plate modeling. Meanwhile, it corresponds to the constant strain triangle element and is able to reproduce a simple piecewise constant curvature field. Thus a unified numerical implementation for thin plate analysis can be conveniently realized using the triangular elements with discretization flexibility. The curvature smoothing operation is further introduced to improve the accuracy of the quadratic Hermite triangular element. The effectiveness of the proposed methodology is demonstrated through numerical examples.     

Complete quadratic isoparametric finite elements in fracture mechanics analysis

For about a decade, each and every researcher who used isoparametric quadratic elements to solve fracture mechanics problems employed only incomplete versions of these elements. Although complete formulations of isoparametric elements are almost as available as incomplete ones in general-purpose finite element computer programs, no attempt to use complete formulations has been seen yet. The purpose of this paper is to show how the complete quadratic isoparametric elements can be employed in the field of fracture mechanics.     

Diffraction and refraction of surface waves using finite and infinite elements

The wave problem is introduced and a derivation of Berkhoff's surface wave theory is outlined. Appropriate boundary conditions are described, for finite and infinite boundaries. These equations are then presented in a variational form, which is used as a basis for finite and infinite elements. The elements are used to solve a wide range of unbounded surface wave problems. Comparisons are given with other methods. It is concluded that infinite elements are a competitive method for the solution of such problems.     

Isoparametric Hermite elements

Isoparametric Hermite elements are created using Bogner–Fox–Schmit rectangles on a reference domain and mapping these numerically onto the computational domain. The difficulties involved in devising explicit C¹ shape functions for isoparametric elements are thus avoided, and the resulting elements have all the benefits of full C¹ continuity, the simplicity of the Bogner–Fox–Schmit element and the geometrical flexibility expected from higher-order isoparametric elements. The numerical mapping consists in the finite element solution of a linear boundary value problem, which is inexpensive and is carried out as a preprocessing operation—the required derivatives of the mapping then being supplied to the main analysis as data. Some care is required in defining the differential boundary conditions, and guidance on this is provided. Examples are given showing the success of the mapping procedure, and the use of the resulting elements in the solution of some boundary value problems. The numerical results confirm a convergence analysis provided for the new isoparametric Hermite element.     

Three-dimensional static and dynamic analysis of functionally graded elliptical plates, employing graded finite elements

On the basis of the three-dimensional theory of elasticity, a graded finite element method capable of modeling static and dynamic behaviors of elliptical plates made of functionally graded materials (FGMs) subjected to uniform pressure is developed. In the present paper, two different material properties distributions are considered. For the dynamic analysis, the effective through-the-thickness continuous material properties distribution of the FGM (which is assumed to be composed of ceramic and metallic constituents) is determined based on Mori–Tanaka homogenization technique. The three-dimensional graded finite element formulation is derived based on the principle of minimum potential energy and Rayleigh Ritz method. To solve the time-dependent equations, Newmark’s direct integration method is employed. To present the efficiency of the present work, several numerical examples are included. Since similar results are not available in the literature, results of the present formulations are verified by comparing them with available ones of a homogenous elliptical plate.     

Axisymmetric vibration of infinite piezoelectric cylinders using one-dimensional finite elements

A general finite-element analysis for infinite piezoelectric cylinders has been formulated. The classical three-dimensional elasticity equations of motion are used. The dependence on theta, z, and time are included by assuming appropriate trigonometric functions, and the three-dimensional problem is reduced to a one-dimensional finite element with four degrees of freedom per node. The tabulated results are limited to cylinders with stress-free, shorted electrode (phi=0) boundary conditions at the outside surface of the cylinder. However, solutions for a variety of boundary conditions are possible. Solutions for the piezoelectric cylinder compare favorably with the existing literature. The motion of piezoelectric cylinders with thin coatings is analyzed by modeling the cylinder and thin coating as a layered cylinder.     

基于梯度有限元的异质材料实体优化设计

针对具有空间分布梯度的异质材料实体的优化设计,建立了两相材料梯度有限元的概念,利用拉格朗日单元的形函数对体积分数进行插值,在节点邻域内引入设计变量自适应下界进行梯度控制,利用移动渐近线算法求解优化设计数学模型以使结构满足特定的功能和目标,以金属夹钳为算例验证了该方法的可行性和鲁棒性.     

Vibration of infinite piezoelectric cylinders with anisotropic properties using cylindrical finite elements

Natural frequencies for free vibration of infinite piezoelectric cylinders are computed using finite elements that are formulated in cylindrical coordinates. The finite-element method is used to model the cross-section of the cylinder in r, theta coordinates using circular sectors. Material constants that are functions of theta are allowed to vary in each circular sector and are computed using standard tensor transformations. The accuracy of the finite-element formulation is verified using previous results for isotropic cylinders and axisymmetric piezoelectric cylinders. New results are tabulated for frequencies of free vibration of solid and hollow piezoelectric cylinders of LiNbO(3) of crystal class 3m. Displacements for typical mode shapes are illustrated graphically.     

Improved quadratic elements

A general method to formulate improved quadratic elements is presented. Derivation of the method is based on more accurate shape functions that take into account effects of governing differential equations. These new shape functions have the same form as those of the standard eight-node quadratic element. Therefore, they may be easily adapted into existing programs. The new quadratic element is compared with both standard and standard condensed quadratic elements. To show relative merits of different quadratic elements eigenvalue tests are performed. Several examples ranging from field problems, plane stress and bar vibration are used to demonstrate the applicability of this approach.     

Diffraction of short waves modelled using new mapped wave envelope finite and infinite elements

We consider a two‐dimensional wave diffraction problem from a closed body such that the complex progressive wave potential satisfies the Sommerfield condition and the Helmholtz equation. We are interested in the case where the wavelength is much smaller than any other length dimensions of the problem. We introduce new mapped wave envelope infinite elements to model the potential in the far field, and test them for some simple Dirichlet boundary condition problems. They are used in conjuction with wave envelope finite elements developed earlier [1] to model the potential in the near field. An iterative procedure is used in which an initial estimate of the phase is iteratively improved. The iteration scheme, by which the wave envelope and phase are recovered, is described in detail. Copyright © 1999 John Wiley & Sons, Ltd.     

Incompressible and locking-free finite elements from Rayleigh mode vectors: quadratic polynomial displacement fields

Under pure bending, with an arbitrary patch of plane four-node finite elements, the exact analytical algebraic expressions of deformation, strain and stress fields are numerically captured by a computer algebra program for both compressible and incompressible continua. Linear combinations of Rayleigh displacement vectors yield the Ritz test functions. These coupled fields model pure bending of an Euler-Bernoulli beam with appropriate linearly varying axial strains devoid of shear. Such Courant admissible functions allow an undeformed straight side to curve in flexure. Since these displacement vectors satisfy equilibrium conditions, they are necessarily functions of the Poisson’s ratio. Applications in bio-, micro- and nano-mechanics motivated this formulation that blurs the frontier between the finite and the boundary element methods. Exact integration yields the element stiffness matrix of a compressible convex or concave quadrilateral, or a triangular element with a side node. For the generic energy density integral, the paper furnishes an analytical expression that can be incorporated in Fortran or C ⁺⁺. In isochoric plane strain problems, the Rayleigh kinematic mode of dilatation is replaced by a constant element pressure. The equivalent nodal loadings are calculated according to the Ritz variational statement. Subsequently, without assembling the global stiffness matrix, nodal compatibility and equilibrium equations are solved in terms of Rayleigh modal participation factors.     

Elastic behavior of glass-like functionally graded infinite hollow cylinder under hydrostatic loads using finite element method

A glass-like (viscoelastic) functionally graded cylinder is studied by using finite element method to investigate the mechanical responses. A subroutine is developed by using ANSYS parametric design language (APDL) to simulate two nonlinearities, which are the variation of material properties with respect to time and position. The cylinder is made of two different viscoelastic materials, namely, pure material one at inner and pure material two at outer surfaces. The material properties are assumed to be presented by simple power law distribution and moreover, bulk and shear moduli are varying with respect to time using the kernel functions depicted regarding Prony series. It is shown that the hoop stresses take the same values at the mean radius (middle of the thickness) for different values of time and grading index. It is found that the radial stress decreases to certain values for specific grading index and then by increasing the grading index it increases to maximum value that related to pure material cylinder. It is shown that unlike the zero axial stress in pure material cylinders, it varies along the thickness from minimum to maximum at inner and outer surfaces, respectively. It is concluded that the viscoelastic functionally graded (VFG) materials play an important role in steady and transient response of hollow cylinder under hydrostatic load.     

More on infinite elements

The method of infinite elements is briefly reviewed. New more logical and general formulations of infinite elements are presented. The simplicity of the programming is emphasized. Results are given for elasticity and potential problems. Future extensions are discussed.