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 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).     

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 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 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.     

Formulation and convergence of a mixed finite element method applied to elastic arches of arbitrary geometry and loading

A sophisticated finite element for elastic arches of arbitrary geometry and loading is developed. The proposed finite element model is based on a mixed variational principle. Both stresses and displacements are obtained as primary results of analysis. Convergence of the method is proven and rates of convergence for stresses and displacements are established. Numerical examples are included demonstrating rapid convergence, and confirm its rate.     

A finite element model for delamination propagation in composites

In this paper the theory and application of a modelling technique for three-dimensional planar delamination growth in laminated composites has been presented. The method is based on linear elastic fracture mechanics assumptions for delamination cracks and uses a strain energy release rate criterion. For a given component, strain energy release rate is considered to be a non-linear function of the location of the delamination front. Hence, satisfaction of the growth criterion reduces to the solution of a non-linear system of equations. A generalized secant method, by the Broyden's update method, is used to solve the system of non-linear equations. Several examples of the application of the technique are presented.     

A finite element creep crack growth model for Waspaloy

Plastic and creep deformations lead to reduced stress levels ahead of the crack tip in a creep crack growth test. However, they can also cause microcracks, cavities and other defects forcing fracture. Numerous damage models are reported in the literature to describe the behavior. In this article, a damage model will be developed from different theories and will be used to describe the creep crack growth behavior of Waspaloy at 973 K. Material parameters for this model are adjusted to uniaxial creep and tensile tests. The calculated creep crack growth curves match very well with the experimental ones supporting the model.     

A 3D finite element model for free-surface flows

The present work deals with the validation of 3D finite element model for free-surface flows. The model uses the non-hydrostatic pressure and the eddy viscosities from the conventional linear turbulence model are modified to account for the secondary effects generated by strong channel curvature in the natural rivers with meandering open channels. The unsteady Reynolds-averaged Navier–Stokes equations are solved on the unstructured grid using the Raviart–Thomas finite element for the horizontal velocity components, and the common P1 linear finite element in the vertical direction. To provide the accurate resolution at the bed and the free-surface, the governing equations are solved in the multi-layers system (the vertical plane of the domain is subdivided into fixed thickness layers). The up-to-date k–ε turbulence solver is implemented for computing eddy coefficients, the Eulerian–Lagrangian–Galerkin (ELG) temporal scheme is performed for enhancing numerical time integration to guarantee high degree of mass conservation while the CFL restriction is eliminated. The present paper reports on successful validation of the numerical model through available benchmark tests with increasing complexity, using the high quality and high spatial resolution three-dimensional data set collected from experiments.     

A wave equation model for finite element tidal computations

A shallow water wave equation is developed from the primitive two-dimensional shallow water equation. A finite element model based on this equation and the primitive momentum equation is developed. A finite difference formulation is used in the time domain which allows the model to be implicit or explicit while still centered in time. Results obtained with linear triangles and quadratic quadrilaterals are reported, and compare well with analytic solutions. The model incorporates all of the economical advantages of earlier models, and errors due to short wavelength spatial noise are suppressed without recourse to artificial means.     

A high-continuity finite element model for two-dimensional elastic problems

A finite element model for the analysis of two-dimensional elastic problems is presented. The proposed discretization is based on a biquadratic interpolation for the displacement components and takes advantage of the enforcement of the interelement continuity to obtain a profitable reduction of the total number of the degrees of freedom. One node (two kinematical parameters) per element only is required.Numerical results obtained for some test problems show the accuracy of the model in analyzing both the deformations and the stress distribution.