Boundary element formulation for two-dimensional creep problems using isoparametric quadratic elements

A boundary element formulation for creep and time-dependent material behaviour problems based on an initial strain approach is presented. The details of numerical algorithm are shown where isoparametric quadratic elements are used both for the boundary elements and the quadrilateral domain cells. The Euler method with automatic time-step control scheme is implemented for time integration. Two creep power laws, time-hardening and strain-hardening, are employed to analyse a number of problems, including a square plate, a plate with a circular hole and a plate with a semi-circular notch subjected to a uniaxial load. The results are compared with analytical solutions where available and the corresponding finite element solutions.     

A stress analysis technique for plate and shell built-up structures with junctions and its application to minimum-weight design of stiffened structures

A finite element formulation using the penalty function method to analyse exactly the junctions of plate and shell built-up structures is suggested for an isoparametric shell element. The connectivity condition at the junction is added to the potential energy functional by the penalty parameter and the interpolating function of displacements. This formulation yields an integral-type stiffness matrix of the special junction elements, which can directly evaluate the surface tractions at the junction. For applying the technique suggested here to the optimum design of structures with junction parts, a design sensitivity analysis formulation for the adhesive special element is also developed. The technique is applied to the minimum-weight design problems of isotropic and composite laminated plates with a stiffener subjected to stress constraints.     

A finite element method for thermoviscoelastic analysis of plane problems

A stress analysis for plane problems in linear thermoviscoelasticity using a finite element formulation is presented. The method employed is based on the assumptions that (1) the material is isotropic, homogeneous and linear, (2) the stress-strain laws are expressed in the hereditary integral form, and (3) the material is thermorheologically simple, which implies that the temperature-time equivalence hypothesis is valid. The associated computer program utilizes isoparametric plane elements.The element matrices that result in the equilibrium equations involve hereditary integrals, and these are approximated by a finite difference scheme for time marching. The solutions for two problems are compared with analytical results evaluated by the integral transform method.For approximate results which require less computer time an alternative form of equilibrium equations utilizing an iterative technique is presented and an example solution is included. Finally, the effect of incompressibility is considered for an axisymmetric numerical example.     

Numerical implementation of a discontinuous finite element algorithm for phase-change problems

This paper is devoted to the numerical solution of phase-change problems in two dimensions. The technique of finite elements is employed. The discretization is carried out using linear isoparametric elements and special attention is given to the accurate integration of functions presenting discontinuities at arbitrarily curved interfaces. This type of function arises in a natural way when dealing with phase-change problems because the enthalpy attains a discontinuity at the phase change temperature. To integrate the discontinuous functions in the phase-changing elements a second mapping is performed from the master element onto a new one for which the interface iis a straight line. The integrals are calculated using the Gaussian technique applied to each part of the divided element, which may be triangular or quadrilateral. The discontinuous integration technique improves the behaviour of the numerical method avoiding any possible loss of latent heat due to an inaccurate evaluation of the residual vector. Some important aspects of the solution of the nonlinear system of equations are discussed and several numerical examples are presented together with the details of the computational implementation of the algorithm.     

Viscoelastic analysis of nearly incompressible solids

The finite element method is well established for the analysis of structures and other field problems. However, its straightforward application for the analysis of nearly incompressible solids yields erratic results. In the present work, an efficient special purpose code for the Viscoelastic Analysis of Nearly Incompressible Solids (VANIS) is developed using isoparametric elements with selective integration procedure, which is a third order Gauss rule for deviatoric response and second order Gauss rule for volumetric response. The software can be effectively employed for the structures with lower Poisson's ratios. VANIS is based on the direct formulation using linear uncoupled thermoviscoelastic theory for the thermorheologically simple materials. The element library consists of 8-noded plane strain, 8-noded axisymmetric solid and 20-noded three dimensional quadratic isoparametric elements. These elements meet all the possible structural idealisation requirements of the solid continua. Experimentally obtained rigidity modulus can be used directly or expressing it in Prony series. The software is tested on a number of problems and gives very accurate results for all the permissible values of the Poisson's ratio.     

A general three-dimensional L-section beam finite element for elastoplastic large deformation analysis

In this paper, the development of a general three-dimensional L-section beam finite element for elastoplastic large deformation analysis is presented. We propose the generalized interpolation scheme for the isoparametric formulation of three-dimensional beam finite elements and the numerical procedure is developed for elastoplastic large deformation analysis. The formulation is general and effective for other thin-walled section beam finite elements. To show the validity of the formulation proposed, a 2-node three-dimensional L-section beam finite element is implemented in an analysis code. As numerical examples, we first perform elastic small and large deformation analyses of a cantilever beam structure subjected to various tip loadings, and elastoplastic large deformation analysis of the same structure under reversed cyclic tip loading. We then analyze the failures of simply supported beam structures of different lengths and slenderness ratios under elastoplastic large deformation. The same problems are solved using refined shell finite element models of the structures. The numerical results of the L-section beam finite element developed here are compared with the solutions obtained using shell finite element analyses. We also discuss the numerical solutions in detail.     

A refined higher-order C° plate bending element

A general finite element formulation for plate bending problem based on a higher-order displacement model and a three-dimensional state of stress and strain is attempted. The theory incorporates linear and quadratic variations of transverse normal strain and transverse shearing strains and stresses respectively through the thickness of the plate. The 9-noded quadrilateral from the family of two dimensional C° continuous isoparametric elements is then introduced and its performance is evaluated for a wide range of plates under uniformly distributed load and with different support conditions and ranging from very thick to extremely thin situations. The effect of full, reduced and selective integration schemes on the final numerical result is examined. The behaviour of this element with the present formulation is seen to be excellent under all the three integration schemes.     

An improved finite element formulation derived from the method of weighted residuals

By replacing the residual with its least squares fit an improvement has been made to the method of weighted residuals. For finite element applications the least squares fit is made on an element basis. The improved formulation is illustrated by a number of examples drawn mainly from fluid flow problems. The new formulation is closely related to the use of reduced integration, and detailed comparisons with exact and reduced integration are presented.     

Efficient design sensitivity derivatives for multi-load case structures as an integrated part of finite element analysis

In most structural optimization problems the accurate calculation of design sensitivity derivatives is required many times during the optimization process. For large structures with multi-load cases the computational costs are sometimes prohibitive. In this paper an approach for incorporating design sensitivity calculations into the finite element analysis of multi-load case structures is presented. A formulation designed to minimize the computational time for the assembled stiffness matrix derivatives is discussed for different element types. The formulation depends on the implicit differentiation method and requires few additional calculations to obtain the design sensitivity derivatives. The approach is developed and implemented to calculate the design sensitivities for continuum and structural isoparametric elements. To demonstrate the accuracy and efficiency of the developed approach some test cases using different structural and continuum element types are presented.     

An isoparametric finite element with decreased sensitivity to midside node location

A modified isoparametric finite element formulation is developed which is insensitive to midside node placement. An eight-node plane element is developed using the modified approach. Numerical results are presented for both the standard serendipity and modified isoparametric eight-node elements. These results demonstrate that decreased sensitivity to midside node placement occur when the modified formulation is used.     

Isoparametric line and transition finite elements with temperature and temperature gradients as primary variables for two dimensional heat conduction

This paper presents isoparametric line and transition finite element formulation for two dimensional heat conduction. The element properties are derived using weak formulation of the Fourier heat conduction equation and the element approximation where nodal temperatures and the nodal temperature gradients are retained as primary variables. The formulation permits linear temperature distribution through the element thickness. Distributed heat flux as well as convective boundaries are permitted on all four faces of the elements. Furthermore, the elements can have internal heat generation as well as orthotropic material properties. The superiority of the formulation in terms of efficiency and accuracy is demonstrated. Numerical examples are presented to illustrate their applications, and a comparison is made with theoretical results.     

A generalized geometrically nonlinear formulation with large rotations for finite elements with rotational degrees of freedoms

A generalized geometrically nonlinear formulation using total Lagrangian approach is presented for the finite elements with translational as well as rotational degrees of freedoms. An important aspect of the formulation presented here is that the restriction on the magnitude of the nodal rotations is eliminated by retaining true nonlinear nodal rotation terms in the definition of the element displacement field and the consistent derivation of the element properties based on this displacement field. The general derivation and the formulation steps are applicable to any element with translational and rotational nodal degrees of freedoms. The specific forms of the formulation for axisymmetric shells, two-dimensional isoparametric beams, curved shells, two-dimensional transition elements and solid-shell transition elements can be easily derived by considering the explicit forms of the nonlinear nodal rotations for the element at hand. The specific forms of this formulation have already been well tested and applied to various two- and three-dimensional elements, the results for some of which are presented here. Currently it is being applied to the three-dimensional isoparametric beam elements.     

A symmetric finite element formulation for the inverse problem of aquifer transmissivity

This paper presents a symmetric isoparametric finite element formulation for the inverse problem of aquifer transmissivity calculation with known piezometric head. An important aspect of the present formulation is that the groundwater flow equation describing the aquifer behavior is transformed into a second-order differential equation by introducing an artificial variable φ. The two-dimensional, line and transition elements derived based on the weak formulation of this transformed equation possess symmetric matrices. In the formulation of the line elements φ and its derivative in η direction are retained as primary variables. This permits modelling of sudden changes in aquifer width. The transition elements provide a natural connecting link between the two-dimensional elements and the line elements. The line elements provide an efficient means of modelling aquifers with unidirectional flow. Numerical examples are given. A comparison of the results obtained here with the Galerkin finite element solution (nonsymmetric formulation) clearly demonstrates the superiority of the formulation presented here.     

On a thin shell element for non-linear analysis, based on the isoparametric concept

This paper discusses a finite element method model for the large displacement, moderate strain analysis of thin shells. The model is based on an ‘adapted’ reference configuration for a displaced element, separating the displacements into rigid body displacements and strain-producing deformations. A strategy is developed, making use of the isoparametric concept for both the choice of reference configuration and in the element formulation. This makes the use of arbitrarily shaped elements possible. The model is shown to give accurate results for a range of relevant problems. Some problems in the general application of this type of model are discussed.     

Nonlinear analysis of an axisymmetric shell using three noded degenerated isoparametric shell elements

An updated Lagrangian formulation of a quadratic degenerated isoparametric shell element is presented for geometrically nonlinear elasto-plastic shell problems. A finite rotation effect is included in the formulation by adopting a co-rotational scheme. The load stiffness matrix has been derived for the treatment of a pressure load. For elasto-plastic behavior, the layered element model is used. The Newton-Raphson iteration method is employed to solve incremental nonlinear equations. For tracking of post-buckling behavior, the work control method is taken into account. Verification of the present technique is obtained by analyzing the available reference problems. Good correlations between the computed results and referenced data can be drawn.     

Cubic isoparametric rolling/traveling finite elements

In this paper, the development and formulation of a new cubic isoparametric rolling/travelling finite element is described. The element can be used to simulate the dyanamic response of steadily rolling/traveling viscoelastic structures without the necessity of numerical time integration of the governing equations. Due to its higher order shape function, the element can more accurately simulate both the inertia and viscoelastic effects associated with rolling/traveling structures than currently available lower order moving elements. To illustrate the capabilities of the new cubic element, comparisons with exact and lower order generated results are presented.     

Collocation and galerkin finite element methods for viscoelastic fluid flow—I. : Description of method and problems with fixed geometry

Collocation and Galerkin finite element methods are developed for viscoelastic fluid flow in a fixed geometry. The collocation methods use Hermite cubic polynomials with a global coordinate transformation to permit irregular geometry. The Galerkin method uses isoparametric elements (transformed element by element) with bilinear polynomials for pressure and quadratic polynomials for velocity. Both methods are applied to two-dimensional flow in planar geometry and the Galerkin method is applied to axisymmetric cylindrical geometries as well. The fluid model is a nonlinear Maxwell model but is limited to small elastic components.

The two methods are applied to several test problems. Entry-length problems test the ability to model pressure singularities are velocity discontinuities. Stick-slip problems test the ability to model pressure singularities and stress discontinuities. Both test problems have analytic or accurate numerical solutions for Newtonian fluids so that the accuracy of the two methods is compared.     

Meshfree-enriched simplex elements with strain smoothing for the finite element analysis of compressible and nearly incompressible solids

This paper presents a meshfree-enriched finite element formulation for triangular and tetrahedral elements in the analysis of two and three-dimensional compressible and nearly incompressible solids. The new formulation is first established in two-dimensional case by introducing a meshfree approximation into a linear triangular finite element with an enriched node. The interpolation functions of the four-noded triangular element are constructed by the meshfree convex approximations and are completed to a polynomial of degree one. The reference mapping using the constructed interpolation functions is shown to be invertible everywhere in the element and the global element area is proven to be conserved under a standard three-point integration rule. The triangular element formulation is extendable to the tetrahedral element in three-dimensional case. To provide a locking-free analysis for the nearly incompressible materials, an area-weighted strain smoothing is developed in conjunction with the enriched interpolation functions to yield a discrete divergence-free property at the integration point. The resultant element formulation with strain smoothing is shown to pass the patch test. To introduce the smoothed strain into Galerkin formulation, a modified Hu–Washizu variational principle is adopted to formulate the discrete equations. Since the Kronecker-delta property in element interpolation is held along the element boundary using meshfree convex approximation, boundary conditions can be treated in a standard way. Several numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed method.     

Integration techniques for isoparametric and higher order bases on finite elements with a curved side

Efficient integration techniques are developed for a class of integrals over finite elements bounded by two straight sides and a parabolic arc. The techniques can be used to speed up the evaluation of the element matrices for both high order transformation bases and for isoparametric bases.     

Design sensitivity analysis with isoparametric shell elements

This paper deals with design sensitivity calculation by the direct differentiation method for isoparametric curved shell elements. Sensitivity parameters include geometric variables which influence the size and the shape of a structure, as well as the shell thickness. The influence of design variables, therefore, may be separated into two distinct contributions. The parametric mapping within an element, as well as the influence of geometric variables on the orientation of an element in space, is accounted for by the sensitivity calculation of geometric variables, and efficient formulations of sensitivity calculation are derived for the element stiffness, the geometric stiffness and the mass matrices. The methods presented here are applied to the sensitivity calculations of displacement, stress, buckling stress and natural frequency of typical basic examples such as a square plate and a cylindrical shell. The numerical results are compared with the theoretical solutions and finite difference values.