A general finite difference method for arbitrary meshes

A two-dimensional finite-difference technique for irregular meshes is formulated for derivatives up to the second order. The domain in the vicinity of a given central point is broken into eight 45 degree pie shaped segments and the closest finite-difference point in each segment to the center point is noted. By utilizing Taylor series expansions about a central point with a unique averaging process for the points in the four diagonal segments, good approximations to all derivatives up to the second order and including the mixed derivatives are obtained. For square meshes the general derivative expressions for arbitrary meshes which were determined reduce to the usual finite difference formulae. In one example problem the Poisson equation is solved for an irregular mesh. In a second example for the first time a problem with a geometric nonlinearity, namely large deflection response of a flat membrane, is solved with an irregular mesh. The solutions compare very favorably with results obtained previously. Some discussion is given on possible approaches for determination of finite difference derivatives higher than the second.     

A finite element formulation for shells of negative Gaussian curvature

An expression for the strain energy of a shell of negative Gaussian curvature, including thickness shear deformations and without neglecting z/R in comparison with unity, is derived. Then a curved trapezoidal finite element formulation based on the principle of minimum potential energy is obtained. The shell element has eight nodes with 40 degrees of freedom and at each node there are three displacements and two rotations. The formulation is applicable for both thin and moderately thick shell analysis. The performance of this finite element is verified by applying it to some problems existing in the literature.     

Least-squares finite element formulation for shear-deformable shells

We present a least-squares based finite element formulation for the numerical analysis of shear-deformable shell structures. The variational problem is obtained by minimizing the least-squares functional, defined as the sum of the squares of the shell equilibrium equations residuals measured in suitable norms of Hilbert spaces. The use of least-squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise, i.e. stability requirements such as inf–sup conditions never arise. The proposed formulation retains the generalized displacements and stress resultants as independent variables and, in view of the nature of the variational setting upon which the finite element model is built, allows for equal-order interpolation. A p-type hierarchical basis is used to construct the discrete finite element model based on the least-squares formulation. Exponentially fast decay of the least-squares functional is verified for increasing order of the modal expansions. Several well established benchmark problems are solved to demonstrate the predictive capability of the least-squares based shell elements. Shell elements based on this formulation are shown to be effective in both membrane- and bending-dominated states.     

A semi-analytical coupled finite element formulation for shells conveying fluids

A new formulation, based on the semi-analytical finite element method, is proposed for elastic shells conveying fluids. The structural equations are based on the shell element proposed by Ramasamy and Ganesan [Comput Struct 70 (1998) 363] while the fluid model is based on velocity potential. Dynamic pressure acting on the walls is derived from Bernoulli's equation. Imposing the requirement that the normal components of velocity of the solid and fluid be equal, introduces fluid–structure coupling. The proposed technique has been validated using results available in the literature. This study shows that instability occurs at a critical fluid velocity corresponding to the shell circumferential mode with the lowest natural frequency and this phenomenon is also independent of the type of structural boundary conditions imposed.     

A nine-node assumed-strain finite element for composite plates and shells

A new finite element for modeling fiber-reinforced composite plates and shells is developed and its performance for static linear problems is evaluated. The element is a nine-node degenerate solid shell element based on a modified Hellinger-Reissner principle with independent inplane and transverse shear strains. Several numerical examples are solved and the solutions are compared with other available finite solutions and with classical lamination theory. The results show that the present element yields accurate solutions for the test problems presented. Convergence characteristics are good, and the solution is relatively insensitive in element distortion. The element is also shown to be free of locking even for thin laminates.     

A geometrical and physical nonlinear finite element model for spatial thin-walled beams with arbitrary section

Based on the theories of Bernoulli-Euler beams and Vlasov's thin-walled members,a new geometrical and physical nonlinear beam element model is developed by applying an interior node in the element and independent interpolations on bending angles and warp,in which factors such as traverse shear deformation,torsional shear deformation and their coupling,coupling of flexure and torsion,and second shear stress are all considered.Thereafter,geometrical nonlinear strain in total Lagarange(TL) and the correspondin...     

Variational geometry: A new method for modifying part geometry for finite element analysis

Interactive graphic methods have the potential to significantly reduce the cost associated with pre- and post-processing of finite element analyses. One area of particular importance is the creation and modification of part geometry.

This paper describes a powerful method for modification of geometry for finite element analysis pre-processors. The method, called “Variational Geometry”, uses a single representation to describe the entire family of geometries that share a generic shape.

A solid geometric model of a component is defined with respect to a set of scalar parameters. Dimensions, such as those which appear on a mechanical drawing, are treated as constraints on the permissible values of these parameters. Constraints on the geometry are expressed as a set of non-linear algebraic equations. The values of the parameters and hence the geometry may be determined by solving the set of non-linear constraint equations.

A procedure for minimizing the computational requirements is presented. For a part with n degrees of freedom, the solution time is shown to be O(n).     

Stochastic finite element analysis of shells

In the present paper the stochastic formulation of the triangular composite (TRIC) facet shell element is presented using the weighted integral and local average methods. The elastic modulus of the structure is considered to be a two-dimensional homogeneous stochastic field which is represented via the spectral representation method. As a result of the proposed derivation and the special features of the element, the stochastic stiffness matrix is calculated in terms of a minimum number of random variables of the stochastic field giving a cost-effective stochastic matrix. Under the assumption of a pre-specified power spectral density function of the stochastic field, it is possible to compute the response variability of the shell structure. Numerical tests are provided to demonstrate the applicability of the proposed methodologies.     

A finite element method for investigating general elastic multi-structures

A finite element method is proposed for investigating the general elastic multi-structure problem, where displacements on bodies, longitudinal displacements on plates, longitudinal displacements and rotational angles on rods are discretized using conforming linear elements, transverse displacements on plates and rods are discretized respectively using TRUNC elements and Hermite elements of third order, and the discrete generalized displacement fields in individual elastic members are coupled together by some feasible interface conditions. The unique solvability of the method is verified by the Lax–Milgram lemma after deriving generalized Korn’s inequalities in some nonconforming element spaces on elastic multi-structures. The quasi-optimal error estimate in the energy norm is also established. Some numerical results are presented at the end.     

A finite element model for thick beams

A general method for obtaining higher order beam theories is reviewed and cast in a form for creating a finite element model. Reissner's principle and Legendre polynomial series expansions are key features in the development. A thick beam element is produced having capabilities of representing nonlinear distributions, through the thickness, of all stress and deformation variables. The model can be used to analyze most thick beams and localized stress conditions. Beam problems are solved and the performance of the thick beam element model is assessed.     

A hybrid strain technique for finite element analysis of plates and shells

A specialization of the Hu-Washizu [1] functional wherein strains and displacements are taken as independent variables is employed in the formulation of ‘hybrid’ elements. Both the strains and displacements are independently interpolated with the strains being eliminated at the element level, leaving displacement variables only to be assembled into the global system of equations. This distinguishes such elements as ‘hybrid’, in contrast to ‘mixed’ wherein the global system of equations contains all the discretized variables. Applications including ‘thick’ plate and shell elements are considered. In many applications the hybrid strain technique appears more natural than the hybrid stress technique since stress discontinuities are accommodated quite conveniently.     

High-precision finite element analysis of cylindrical shells

This paper presents the finite element formulation to study the free vibration of cylindrical shells. The displacement function for the high-precision shell element with 16 degrees of freedom is approximated by a Hermitian polynomial of beam function type. The explicit formulation for the high-precision element is extremely efficient. For the purpose of comparison, the subject element is used to study the sample case of free vibration of a shell structure. The results are in good agreement with those published. The study shows that solution accuracy with fewer elements is assured and that accurate solutions are obtainable in the high-frequency range.     

An arbitrary lagrangian-eulerian finite element method for path-dependent materials

The conservation laws, the constitutive equations, and the equation of state for path-dependent materials are formulated for an arbitrary Lagrangian-Eulerian finite element method. Both the geometrical and material nonlinearities are included in this setting. Computer implementations are presented and an elastic-plastic wave propagation problem is used to examine some features of the proposed method.     

A three-dimensional state space finite element solution for laminated composite cylindrical shells

On the basis of the theory of three-dimensional elasticity, this paper presents a state space finite element solution for stress analysis of cross-ply laminated composite shells. This is a continuation of the authors’ previously published work on laminated plates [Compos. Struct. 57 (1–4) (2002) 117; Comput. Methods Appl. Mech. Engrg. 191 (37–38) (2002) 4259]. Once again a state space formulation is introduced to solve for through-thickness stress distributions, while the traditional finite elements are used to approximate the in-surface variations of state variables. A three-dimensional laminated shell element is established in an arbitrary orthogonal curvilinear coordinate system, while the application of the element is shown by calculating stresses in laminated cylindrical shells. Compared with the traditional finite element method, the new solution provides accurate continuous through-thickness distributions of both displacements and transverse stresses.     

A finite element model for martensitic thin films

Abstract A finite element approximation of the thin film limit for a sharp interface bulk energy for martensitic crystals is given. The energy density models the softening of the elastic modulus controlling the low-energy path from the cubic to the tetragonal lattice, the loss of stability of the tetragonal phase at high temperatures and the loss of stability of the cubic phase at low temperatures, and the effect of compositional fluctuation on the transformation temperature. The finite element approximation is then used to simulate the hysteresis of a martensitic thin film obtained after applying a biaxial loading cycle to the film below the transformation temperature. Keywords: finite element, phase transformation, martensite, austenite, thin film Mathematics Subject Classification (1991): 65C30, 65Z05, 74K35, 74N10, 74N15, 74S05     

A general isoparametric finite element program SDRC SUPERB

SDRC SUPERB is a general purpose finite element program that performs linear static, dynamic and steady state heat conduction analyses of structures made of isotropic and/or orthotropic elastic materials having temperature dependent properties. The finite element library of SUPERB contains isoparametric plane stress, plane strain, flat plate, curved shell, solid type curved shell and solid elements in addition to conventional beam and spring elements. Linear, quadratic and cubic interpolation functions are available for all isoparametric elements. Independent parameters such as displacements and temperatures are obtained from SUPERB using the stiffness method of analysis. The remaining dependent parameters, such as stresses and strains, are evaluated at element gauss points and extrapolated to nodal locations. Averaged values are given as final output. The graphic capabilities of SUPERB consists of geometry and distorted geometry plotting, and stress, strain and temperature contouring. Contours are plotted at user defined cutting planes for solids and at top, middle or bottom surfaces for plate and shell types of structures.In the first part of this paper, the program capabilities of SUPERB are summarized. Extrapolation techniques used for determining dependent nodal parameters and for contour plotting are explained in the second part of the paper. Behavior of standard, wedge and transition type isoparametric elements and the effect of interpolation function orders on accuracy are discussed in the third part. The results of several illustrative problems are included.     

Quasi-analytical finite element procedures for axisymmetric anisotropic shells and solids

Using complex series representations, a quasi-analytical finite element procedure is developed which can analyze the static and dynamic mechanical fields of anisotropic axisymmetric shells and bodies. Due to its generality the procedure can handle arbitrary laminate construction with possible meridional and radial variations in locally or globally mechanically anisotropic materials. In this respect, in contrast to traditional quasi-analytical procedures which are limited to the ‘specially’ orthotropic case, the present treatment reveals several important effects of material and/or structural anisotropy. To illustrate the procedure as well as the significant effects of material anisotropy, several numerical examples are given along with comparisons with known analytical treatments.     

Nonlinear finite element analysis of shells: Part I. three-dimensional shells

A nonlinear finite element formulation is presented for the three-dimensional quasistatic analysis of shells which accounts for large strain and rotation effects, and accommodates a fairly general class of nonlinear, finite-deformation constitutive equations. Several features of the developments are noteworthy, namely: the extension of the selective integration procedure to the general nonlinear case which, in particular, facilitates the development of a ‘heterosis-type’ nonlinear shell element; the presentation of a nonlinear constitutive algorithm which is ‘incrementally objective’ for large rotation increments, and maintains the zero normal-stress condition in the rotating stress coordinate system; and a simple treatment of finite-rotational nodal degrees-of-freedom which precludes the appearance of zero-energy in-plane rotational modes. Numerical results indicate the good behavior of the elements studied.