Showing papers in "International Journal for Numerical Methods in Engineering in 1984"

A Taylor–Galerkin method for convective transport problems

Abstract: A method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions. To produce accurate temporal differencing, the method employs forward-time Taylor series expansions including time derivatives of second- and third-order which are evaluated from the governing partial differential equation. This yields a generalized time-discretized equation which is successively discretized in space by means of the standard Bubnov–Galerkin finite element method. The technique is illustrated first in one space dimension. With linear elements and Euler, leap-frog and Crank–Nicolson time stepping, several interesting relations with standard Galerkin and recently developed Petrov–Galerkin methods emerge and the new Taylor–Galerkin schemes are found to exhibit particularly high phase-accuracy with minimal numerical damping. The method is successively extended to deal with variable coefficient problems and multi-dimensional situations.

Rational approach for assumed stress finite elements

Abstract: A new method for the formulation of hybrid elements by the Hellinger-Reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element. The equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co-ordinates. The resulting element possesses all the ideal qualities, i.e. it is invariant, it is less sensitive to geometric distortion, it contains a minimum number of stress parameters and it provides accurate stress calculations. For the formulation of a 4-node plane stress element, a small perturbation method is used to determine the equilibrium constraint equations. The element has been proved to be always rank sufficient.

Automatic three‐dimensional mesh generation by the finite octree technique

Abstract: An octree-based fully automatic three-dimensional mesh generator is presented. The mesh generator is capable of meshing non-manifold models of arbitrary geometric complexity through the explicit tracking and enforcement of geometric compatibility and geometric similarity at each step of the meshing process. The resulting procedure demonstrates a linear growth rate with respect to the number of elements and can be easily integrated with any geometric modeller through a set of geometric operators.

On a certain mixed variational theorem and a proposed application

Abstract: Formulation d'une equation variationnelle pour les deplacements et certaines contraintes en vue d'une application aux plaques elastiques stratifiees isotropes et anisotropes

Algorithms for refining triangular grids suitable for adaptive and multigrid techniques

Abstract: Two general algorithms for refining triangular computational meshes based on the bisection of triangles by the longest side are presented and discussed. The algorithms can be applied globally or locally for selective refinement of any conforming triangulation and always generate a new conforming triangulation after a finite number of interactions even when locally used. The algorithms also ensure that all angles in subsequent refined triangulations are greater than or equal to half the smallest angle in the original triangulation; the shape regularity of all triangles is maintained and the transition between small and large triangles is smooth in a natural way. Proofs of the above properties are presented. The second algorithm is a simpler, improved version of the first which retains most of the properties of the latter. The algorithms can be used either for constructing irregular computational meshes or for locally refining any given triangulation. In this sense they can be adequately combined with adaptive and/or multigrid techniques for solving finite element systems. Examples of the application of the algorithms are given and two possible generalizations are pointed out.

The post-processing approach in the finite element method—part 1: Calculation of displacements, stresses and other higher derivatives of the displacements

Abstract: This is the first in a series of three papers in which we discuss a method for ‘post-processing’ a finite element solution to obtain high accuracy approximations for displacements, stresses, stress intensity factors, etc. Rather than take the values of these quantities ‘directly’ from the finite element solution, we evaluate certain weighted averages of the solution over the entire region. These yield approximations are of the same order of accuracy as the strain energy. We obtain error estimates, and also present some numerical examples to illustrate the practical effectiveness of the technique. In the third paper of this series we address the matters of adaptive mesh selection and a posteriori error estimation.

The post‐processing approach in the finite element method—Part 2: The calculation of stress intensity factors

Abstract: : In the context of a model problem the authors describe post-processing techniques for the calculation of generalized stress intensity factors. They discuss two broad classes of methods, one involving an influence function, and the other related to the well-known energy release principle of fracture mechanics. An error analysis is sketched and two numerical examples are given to illustrate the effectivity of the techniques. This is the second in a series of three papers. (Author)

A unified set of single step algorithms. Part 1: General formulation and applications

Abstract: A general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented. This algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities. In many cases it presents a computationally advantageous form over conventional procedures—this is particularly so when compared with the Newmark algorighm and its variants.

Finite elements with increased freedom in choosing shape functions

Abstract: Starting with a mathematical statement of the convergence requirements for an element stiffness matrix, the paper discusses displacement shape functions that may be used in connection with the potential energy principle. In short, these functions must be force orthogonal and energy orthogonal, but they need not be conforming (satisfy interelement compatibility). It is shown that the requirements to the displacement functions may be greatly relaxed through slight modifications of the coupling stiffness between fundamental and higher order displacement modes. Several alternative formulations are examined. In particular, a new ‘free formulation’ is suggested. Using this form, which is very simple, the only requirement to the displacement patterns used is that they should contain the fundamental deformation modes and be linearly independent. Applications of the theory to triangular and rectangular plate bending elements are shown; the simple stiffness matrix for the latter is given explicitly. The numerical results compare favourably with other types of finite elements.

Adaptive Eulerian–Lagrangian finite element method for advection–dispersion

Abstract: A new adaptive finite element method is proposed for the advection–dispersion equation using an Eulerian–Lagrangian formulation. The method is based on a decomposition of the concentration field into two parts, one advective and one dispersive, in a rigorous manner that does not leave room for ambiguity. The advective component of steep concentration fronts is tracked forward with the aid of moving particles clustered around each front. Away from such fronts the advection problem is handled by an efficient modified method of characteristics called single-step reverse particle tracking. When a front dissipates with time, its forward tracking stops automatically and the corresponding cloud of particles is eliminated. The dispersion problem is solved by an unconventional Lagrangian finite element formulation on a fixed grid which involves only symmetric and diagonal matrices. Preliminary tests against analytical solutions of one- and two-dimensional dispersion in a uniform steady-state velocity field suggest that the proposed adaptive method can handle the entire range of Peclet numbers from 0 to ∞, with Courant numbers well in excess of 1.

The Post-Processing Approach in the Finite Element Method. Part 3. A Posteriori Error Estimates and Adaptive Mesh Selection.

Abstract: This paper is the final in a series of three in which we have discussed a finite element post-processing technique. Here we shall deal with the questions of adaptive mesh selection and a posteriori error estimation. Some numerical examples computed by the FEARS program will be used to illustrate the approaches taken.

The self-equilibration of residuals and complementarya posteriori error estimates in the finite element method

Abstract: It is demonstrated that the residual in a compatible (displacement) finite element solution can be partitioned into local self-equilibrating systems on each element. An a posteriori error analysis is then based on a complementary approach and examples indicate that the guaranteed upper bound on the energy of the error is preserved.

Path independent integrals for computing stress intensity factors at sharp notches in elastic plates

Abstract: A set of path independent integrals is constructed for the calculation of the generalized stress intensity factors occurring in elastic plates having sharp re-entrant corners or notches with stress-free faces and subjected to Mode I, II or III type loading. The Mode I integral is then demonstrated to enjoy a reasonable degree of numerical path independence in a finite element analysis of a test problem having an exact solution. Finally, this integral is used on the same problem in conjunction with a regularizing, finite element, procedure or superposition method. The results indicate that sufficiently accurate estimates of these stress intensity factors for engineering purposes can be achieved with little computational effort.

On the use of quarter‐point boundary elements for stress intensity factor computations

Abstract: This communication studies a procedure for stress intensity factor computations using traction singular quarter-point boundary elements. Opening mode stress intensity factors are computed from the tractions' nodal values at the crack tip. A comparison is made between the factors calculated using this procedure and those obtained by previously recommended methods which made use of the nodal values of the displacements. The proposed procedure was seen to be less discretization sensitive than any other of the considered methods. Accurate results were obtained even in the case of coarse meshes.

Boundary integral equation method for linear porous‐elasticity with applications to soil consolidation

Abstract: For physical phenomena governed by the Biot model of porous-elasticity, a reciprocal relation, similar to the Betti's recoprocal theorem in elasticity, is constructed in Laplace transformed space. Integrating the reciprocal relation enables one to formulate boundary integral equations. The fundamental kernels for the integral equations are solved in closed forms for the case of isotropic material. Numerical implementation of two-dimensional problems includes finite element ideas of discretization and polynomial interpolation, and numerical inversion of a Laplace transform. Practical applications of the method are found in consolidation problems in soils which contain compressible as well as incompressible pore fluids. Also, as a numerical experiment, consolidation of partially saturated soil is simulated and interesting phenomena are observed. The currently developed boundary integral equation method (BIEM) for porous-elasticity may be viewed as an efficient and accurate alternative of existing finite element and finite difference methods. For linear consolidation problems, application of BIEM is always preferred to the other numerical methods whenever possible.

A C0 triangular plate element with one‐point quadrature

Abstract: On developpe un element de plaque triangulaire du type de Mindlin qui utilise des champs lineaires pour les rotations et les deflexions transversales. L'element ne necessite qu'un seul point de quadrature ce qui est tres interessant pour l'analyse non lineaire

Analysis of laminated composite shells using a degenerated 3‐D element

Abstract: A special three-dimensional element based on the total Lagrangian description of the motion of a layered anisotropic composite medium is developed, validated and employed to analyze laminated anisotropic composite shells. The element contains the following features: geometric nonlinearity, dynamic (transient) behavior and arbitrary lamination scheme and lamina properties. Numerical results of nonlinear bending, natural vibration, and transient response are presented to illustrate the capabilities of the element.

Efficient linear and nonlinear heat conduction with a quadrilateral element

Abstract: A method is presented for performing efficient and stable finite element calculations of heat conduction with quadrilaterals using one-point quadrature. The stability in space is obtained by using a stabilization matrix which is orthogonal to all linear fields and its magnitude is determined by a stabilization parameter. It is shown that the accuracy is almost independent of the value of the stabilization parameter over a wide range of values; in fact, the values 3, 2 and 1 for the normalized stabilization parameter lead to the 5-point finite difference, 9-point finite difference and fully integrated finite element operators, respectively, for rectangular meshes; numerical experiments reported here show that the three have identical rates of convergence in the L2 norm. Eigenvalues of the element matrices, which are needed for stability limits, are also given. Numerical applications are used to show that the method yields accurate solutions with large increases in efficiency, particularly in nonlinear problems.

Finite layer analysis of layered elastic materials using a flexibility approach. Part 1—Strip loadings

Abstract: It is well known that the analysis of a horizontally layered elastic material can be considerably simplified by the introduction of a Fourier or Hankel transform and the application of the finite layer approach. The conventional finite layer (and finite element) stiffness approach breaks down when applied to incompressible materials. In this paper these difficulties are overcome by the introduction of an exact finite layer flexibility matrix. This flexibility matrix can be assembled in much the same way as the stiffness matrix and does not suffer from the disadvantage of becoming infinite for an incompressible material. The method is illustrated by a series of examples drawn from the geotechnical area, where it is observed that many natural and man-made deposits are horizontally layered and where it is necessary to consider incompressible behaviour for undrained conditions. For abstract of part 2 see TRIS no. 378330. (Author/TRRL)

An efficient zooming method for finite element analysis

Abstract: In this paper, an efficient, exact zooming technique is developed which employs static condensation and exact structural reanalysis methods. For a multiple level of zooming such that the successive level of zooming is contained within the prior levels of zoom, repeated application of static condensation will reduce the system to one that is associated only with the degrees-of-freedom (dof) of the original model. Then, application of an exact static reanalysis technique permits the displacements at the dof of the original model that are contained in the final zoomed portion of the structure to be obtained first. Next, the response external to the zoom, as well as the response of additional dof within various levels of zooming, can be computed. With the triangular factor of the stiffness matrix of the original system available, this approach involves only the solution of a system of equations of small order. The proposed method is demonstrated by a numerical example.

A recursive quadratic programming method with active set strategy for optimal design

Abstract: Recursive quadratic programming methods have become popular in the field of mathematical programming owing to their excellent convergence characteristics. There are two recursive quadratic programming methods that have been published in the literature. One is by Han and the other is by Pshenichny, published in 1977 and 1970, respectively. The algorithm of Pshenichny had been undiscovered until now, and is examined here for the first time. It is found that the proof of global convergence by Han requires computing sensitivity coefficients (derivatives) of all constraint functions of the problem at every iteration. This is prohibitively expensive for large-scale applications in optimal design. In contrast, Pshenichny has proved global convergence of his algorithm using only an active-set strategy. This is clearly preferable for large-scale applications. The method of Pshenichny has been coded into a FORTRAN program. Applications of this method to four example problems are presented. The method is found to be very reliable. However, the method is found to be very sensitive to local minima, i.e. it converges to a local minimum nearest to the starting design. Thus, for optimal design problems (which usually possess multiple local minima) it is suggested that Pshenichny's method be used as part of a hybrid method.

A total lagrangian formulation for the geometrically nonlinear analysis of structures using finite elements. Part I. Two-dimensional problems: Shell and plate structures

Abstract: A total Lagrangian finite element formulation for the geometrically nonlinear analysis (large displacement/large rotations) of shells is presented. Explicit expressions of all relevant finite element matrices are obtained by means of the definition of a local co-ordinate system, based on the shell principal curvature directions, for the evaluation of strains and stresses. A series of examples of nonlinear analysis of shell and plate structures is given.

A simple element for static and dynamic response of beams with material and geometric nonlinearities

Abstract: A unified simple 6 degrees-of-freedom beam finite element and the associated computational procedures have been developed for the fast and efficient solution of a wide class of static and dynamic response problems of the beam type with material and/or geometrical nonlinearities. The material nonlinearity is treated by including its effect in the governing equations by forming the stiffness matrix of each element using a two-dimensional grid of Gauss points and using the material properties at each point corresponding to the uniaxial strain at that point. Examples are provided for metal and reinforced concrete beams. A powerful yet straightforward method for the solution of elastica problems of beams and frames, using the beam element developed by the senior author, has been extended for determining the dynamic response of beams undergoing large displacements, including large rotations. The solution procedure involves piecewise linearization of response equations and iterations at each incremental step to achieve equilibrium. The solution procedure is simple and easy to apply. A variety of problems is solved to determine the applicability of the proposed simple formulations. Excellent agreement with existing analytical solutions which employ higher order elements demonstrates the efficiency and versatility of the present simple beam element in nonlinear analysis.

Modelization of phase changes by fictitious‐heat flow

Abstract: This paper presents an efficient algorithm to solve phase change problems. The first part gives the definition of the problem and mentions some practical applications. The second part analyses how to deal with phase changes by means of concentrated fictitious heat flows. The principles of the method are described and compared to those of other methods. The concept of ‘the area associated with a node’ is developed and some practical values for the weight coefficients are recommended. Conclusions are given about the convergence and the stability of the algorithm, and about the improvements obtained by prescribing the temperatures and using relaxation techniques. Theoretical and practical examples illustrate the validity and efficiency of the proposed method. Copyright © 1984 John Wiley & Sons, Ltd

A unified set of single step algorithms. Part 2: Theory

Abstract: Part 1 of this paper1 presents a very general single step algorithm SSpj for the numerical integration of first and second order time dependent differential equations. In Part 2 we present and discuss results of the accuracy and stability analysis for SS11, SS21, SS31, SS22 and SS32. There is also a detailed comparison of SS22 and the Newmark algorithm.

Numerical evaluation of the quarter‐point crack tip element

Abstract: This paper attempts to answer two commonly raised questions during the preparation of a finite element mesh, for the linear elastic fracture analysis of cracked structure: how to set up the finite element mesh around the crack tip, and what level of accuracy is to be expected from such a modelling. Two test problems, with known analytical expressions for their stress intensity factors, are analysed by the finite element method using the isoparametric quadratic singular element. The modified parameters were the order of integration, aspect ratio, number of elements surrounding the crack tip, use of transition elements, the singular element length over the total crack length, the symmetry of the mesh around the crack tip. Based on these analyses, a data base is created and various plots produced. The results are interpreted, the accuracy evaluated and recommendations drawn. Contrary to previous reports, it is found that the computed stress intensity factor (SIF) remains within engineering accuracy (10 per cent) throughout a large range of l/a (singular element length over crack length) for problems with a uniform non-singular stress distribution ahead of the crack tip (i.e. double edge notch), and l/a should be less than 0·1 for problems with a non-singular stress gradient (i.e three-point bend). Also, it is found that the best results are achieved by using at least four singular elements around the crack tip, with their internal angles around 45 degrees, and a reduced (2 × 2) numerical integration.

A shear‐flexible triangular finite element model for laminated composite plates

Abstract: Formulation and numerical evaluation of a shear-flexible triangular laminated composite plate finite element is presented in this paper. The element has three nodes at its vertices, and displacements and rotations along with their first derivatives have been chosen as nodal degrees-of-freedom. Computation of element matrices is highly simplified by employing a shape function subroutine, and an optimal numerical integration scheme has been used to improve the performance. The element has satisfactory rate of convergence and acceptable accuracy with mesh refinement for thick as well as thin plates of both homogeneous isotropic and laminated anisotropic materials. The numerical studies also suggest that reliable prediction of the behaviour of laminated composite plates necessitates the use of higher order shear-flexible finite element models, and the proposed finite element appears to have some advantages over available elements.

Analysis of multi-layer laminates using three-dimensional super-elements

Abstract: Analysis of multi-layer composite laminates using three-dimensional finite element models often results in high computational cost. This communication reports a three-dimensional super-element scheme which can reduce the computational cost.