Computation of accurate nodal derivatives of finite element solutions: the finite node displacement method

This paper presents a new method for extracting high‐accuracy nodal derivatives from finite element solutions. The approach involves imposing a finite displacement to individual mesh nodes, and solving a very small problem on the patch of surrounding elements, whose only unknown is the value of the solution at the displaced node. A finite difference between the original and perturbed values provides the directional derivative. Verification is shown for a one‐dimensional diffusion problem with exact nodal solution and for two‐dimensional scalar advective–diffusive problems. For internal nodes the method yields accuracy slightly superior to that of the superconvergent patch recovery (SPR) technique of Zienkiewicz and Zhu (ZZ). We also present a variant of the method to treat boundary nodes. In this case, the local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest. We show that the new method gives normal derivatives at boundary points that are consistent with the so‐called ‘auxiliary fluxes’. The resulting nodal derivatives are much more accurate than those obtained by the ZZ SPR technique. Copyright © 2007 Crown in the right of Canada. Published by John Wiley & Sons, Ltd.     

‘Infinite domain’ elements

A parametric element is formulated which enables the economic modelling of ‘infinite domain’ type problmes. A typical problem is an opening in a stress field in an infinite medium, either in two or three dimensions. The strategy is to model around the opening with two or three layers of conventional isoparametric finite elements and surround these with a single layer of ‘infinite domain’ elements. Several sample problems has been analysed for circular, square and spherical openings in infinite media, and the results compared with either theoretical or boundary element solutions which include the ‘infinite’ boundary in their solution technique.     

Finite element analysis of plane couple-stress problems using first order stress functions

A complementary energy based variational principle, using first order stress functions, is developed for plane linear elastic couple-stress problems. The principle is analogous to that used in a total potential energy based Mindlin/Reissner thick plate bending analysis and as such is a generalization of the classical analogy between plate stretching and plate bending. Traction boundary conditions are enforced using a Lagrange multiplier technique. The resulting C⁰ finite element ‘equilibrium stress model’ is validated by investigating the reduction of the stress concentration factor associated with a small hole in a field of uniform tension.     

Moving particle finite element method with global smoothness

We describe a new version of the moving particle finite element method (MPFEM) that provides solutions within a C⁰ finite element framework. The finite elements determine the weighting for the moving partition of unity. A concept of ‘General Shape Function’ is proposed which extends regular finite element shape functions to a larger domain. These are combined with Shepard functions to obtain a smooth approximation. The Moving Particle Finite Element Method combines desirable features of finite element and meshfree methods. The proposed approach, in fact, can be interpreted as a ‘moving partition of unity finite element method’ or ‘moving kernel finite element method’. This method possesses the robustness and efficiency of the C⁰ finite element method while providing at least C¹ continuity. Copyright © 2004 John Wiley & Sons, Ltd.     

A finite element analysis in polar co-ordinates of the Saint Venant torsion problem

A characteristic feature of the variational functionals for several boundary value problems in polar co-ordinates is the fact that one independent variable occurs explicitly in the denominator. Therefore, the coefficients of the finite element equations for sectors of circular ring shaped elements are not constants but functions of the distance of the elements from the origin of co-ordinates.¹ We name them coefficient functions. In order to show the particular aspects of the calculations in terms of polar co-ordinates we deal here with the solution of the torsion problem by bilinear and bicubic Hermitian interpolation. The finite element equations are arranged according to Schaefer² in the form of block which can easily be transformed into ‘stars’³ or molecules^4,5 similar to those used in finite difference methods. The origin of co-ordinates requires a special consideration, firstly because of the coincidence of several nodes at that point and secondly because of the divergent behaviour of some coefficient functions. It turns out to be advantageous for the numerical calculations to expand the coefficient functions in power series. Besides, the expansions are required to deduce the equations for rectangular elements by limiting processes. The twisting moments and shearing stresses calculated for several cross sections illustrate the numerical suitability of the method. The finite element values are compared partly with exact solutions and partly with experimental results obtained by a moiré method using Prandtl's soap film analogy.⁶ Finally it is shown how the accuracy of the finite element values can be improved by the Richardson extrapolation⁷.     

p-version least-squares finite element formulation of Burgers' equation

A p-version least-squares finite element formulation for non-linear problems is presented and applied to the steady-state, one-dimensional Burgers' equation. The second-order equation is recast as a set of first-order equations which permit the use of C⁰ elements. The primary and auxiliary variables are approximated using equal-order p-version hierarchical approximation functions. The system of non-linear simultaneous algebraic equations resulting from the least-squares process is solved using Newton's method with a line search. The use of ‘exact’ and ‘reduced’ quadrature rules is investigated and the results are compared. The formulation is found to produce excellent results when the ‘exact’ integration rule is used. The combination of least-squares finite element formulation and p-version works extremely well for Burgers' equation and appears to have great potential in fluid dynamics problems.     

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.     

The simple boundary element method for multiple cracks in functionally graded media governed by potential theory: a three‐dimensional Galerkin approach

The simple boundary element method consists of recycling existing codes for homogeneous media to solve problems in non‐homogeneous media while maintaining a purely boundary‐only formulation. Within this scope, this paper presents a ‘simple’ Galerkin boundary element method for multiple cracks in problems governed by potential theory in functionally graded media. Steady‐state heat conduction is investigated for thermal conductivity varying either parabolically, exponentially, or trigonometrically in one or more co‐ordinates. A three‐dimensional implementation which merges the dual boundary integral equation technique with the Galerkin approach is presented. Special emphasis is given to the treatment of crack surfaces and boundary conditions. The test examples simulated with the present method are verified with finite element results using graded finite elements. The numerical examples demonstrate the accuracy and efficiency of the present method especially when multiple interacting cracks are involved. Copyright © 2005 John Wiley & Sons, Ltd.     

Convergence analysis of the Q1-finite element method for elliptic problems with non-boundary-fitted meshes

The aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain's boundary. This problematic of the last century (e.g. the finite difference methodology) returns to topical studies with the huge development of domain embedding, fictitious domain or Cartesian-grid methods. These methods use regular structured meshes (most often Cartesian) for non-aligned domains. Although non-boundary-fitted approaches become more and more applied, very few studies are devoted to theoretical error estimates. In this paper, the convergence of a Q₁-non-conforming finite element method is analyzed for second-order elliptic problems with Dirichlet, Robin or Neumann boundary conditions. The finite element method uses standard Q₁-rectangular finite elements. As the finite element approximate space is not contained in the original solution space, this method is referred to as non-conforming. A stair-step boundary defined from the Cartesian mesh approximates the original domain's boundary. The convergence analysis of the finite element method for such a kind of non-boundary-fitted stair-stepped approximation is not treated in the literature. The study of Dirichlet problems is based on similar techniques as those classically used with boundary-fitted linear triangular finite elements. The estimates obtained for Robin problems are novel and use some more technical arguments. The rate of convergence is proved to be in 𝒪(h^1/2) for the H¹-norm for all general boundary conditions, and classical duality arguments allow one to obtain an 𝒪(h) error estimate in the L²-norm for Dirichlet problems. Numerical results obtained with fictitious domain techniques, which impose original boundary conditions on a non-boundary-fitted approximate immersed interface, are presented. These results confirm the theoretical rates of convergence. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite element methods for an optimal steady-state control problem

An optimal steady-state control problem governed by an elliptic state equation is solved by several finite element methods. Finite element discretizations are applied to different variational formulations of the problem yielding accurate numerical results as compared with the given analytical solution. It is sated that, for minimum computational effort and high accuracy, ‘mixed’ finite elements requiring only C° continuity, and approximating the control and state functions simultaneously are better suited to similar ‘fourth order’ problems.     

Modified flux-vector-based Green element method for problems in steady-state anisotropic media

In this study we present a new numerical technique for solving problems in steady-state heterogeneous anisotropic media, namely the ‘flux-vector-based’ Green element method (‘$\mathit{q}$-based’ GEM) for anisotropic media. This method, which is appropriate for problems where the permeability has either constant or continuous components over the whole domain, is based on the boundary element method (BEM) formulation for direct, steady-state flow problems in anisotropic porous media, which is applied to finite element method (FEM) meshes. For situations involving media discontinuities, an extension of this ‘$\mathit{q}$-based’ GEM formulation is proposed, namely the modified ‘$\mathit{q}$-based’ GEM for anisotropic media. Numerical results are presented for various physical problems that simulate flow in an anisotropic medium with diagonal layers of different permeabilities or around faults and wells, and they show that the new method, with the extensions proposed, is very suitable for steady-state problems in such media.     

Comparison between a finite element and a composite method for a three-dimensional potential problem

This paper is concerned with the solution of the three-dimensional potential problem for electromagnetic river gauging. It extends previous ideas of joining finite elements in an interior region to one infinite external element treated by the boundary integral method^1,2 to this case where there are two external infinite elements representing the river and the ground. The boundary continuity conditions on the infinite river–ground interface, as well as the internal–external interfaces, are dealt with by introducing a variational principle with relaxed continuity requirement³.     

Infinite elements with 1/rn type decay

This paper presents a method of developing a family of 1/rⁿ type infinite elements for the analysis of problems definite in unbounded media. The proposed method is a direct extension of the conventional finite element method. The resulting improper integrals are integrated exactly over the infinite element domains. Two numerical examples in elastic half-space static problems are investigated to illustrate the applicability and accuracy of the method. The use of the proposed infinite elements yields excellent results and preserves all the advantages of the finite element method.     

Finite prism analysis of plates and shells

The finite prism technique introduced by Zienkiewicz and Too⁴ is extended to include 12-node prism elements and, more importantly, a novel offset beam element. The use of 12-node prism elements enables parabolic strain distributions to be simulated, this being useful for structures which have strongly tapered cross-sections. The offset beam element is used to simulate flexure and torsion of a beam whose centroid is offset from the main structure. The element is specified completely at the nodes of adjacent prism elements and so is not really an ‘element’ in the usual sense. The analysis is applied to thin and thick plates and to shells, with and without edge beams. It is shown to be more versatile than the finite strip method and it requires smaller computer resources than the finite element method. Experimental verification of the analysis is obtained by comparison with measurements for a double-T bridge deck tested by Loo¹⁴.     

Green's functions for the Laplace equation in a 3-layer medium,boundary element integrals and their application to cathodic protection

We report in this paper a set of nine Green's functions for the Laplace equation for an infinite 3-layer medium in which a layer of finite width is sandwiched between two semi-infinite domains. Typical 3D plots of these Green's functions are computed and presented. Taking an offshore platform as a prime example of a structure in a 3-layer medium (atmosphere, ocean and soil), we work out the boundary element integrals using macro elements such as the tubulars. Constant elements reduce several of these boundary integrals to analytical forms. As an application, we discuss the cathodic protection modelling of offshore structures using the ‘boundary element method’.     

A simple cubic displacement element for plate bending

Early attempts to construct a triangular finite element for plate bending problems from a compatible cubic displacement field are not entirely satisfactory. The present paper shows how an accurate plate element can be achieved using independent cubic polynomial assumptions for the internal and boundary displacements in conjunction with a modified potential energy principle. This approach yields a simple algebraic formulation with favourable connection quantities at the element vertices which will appeal to practical users of the conventional finite element displacement method. Moreover, in Appendix I it is shown that the cubic element is identical to a previous hybrid stress element with linear internal bending and twisting moments and cubic boundary displacements. The stresses obtained from the former hybrid finite element solution therefore satisfy the strain compatibility conditions exactly. This remarkable result has an important significance in the theory of hybrid finite elements.     

Finite elements for materials with strain gradient effects

A finite element implementation is reported of the Fleck–Hutchinson phenomenological strain gradient theory. This theory fits within the Toupin–Mindlin framework and deals with first‐order strain gradients and the associated work‐conjugate higher‐order stresses. In conventional displacement‐based approaches, the interpolation of displacement requires C¹‐continuity in order to ensure convergence of the finite element procedure for higher‐order theories. Mixed‐type finite elements are developed herein for the Fleck–Hutchinson theory; these elements use standard C⁰‐continuous shape functions and can achieve the same convergence as C¹ elements. These C⁰ elements use displacements and displacement gradients as nodal degrees of freedom. Kinematic constraints between displacement gradients are enforced via the Lagrange multiplier method. The elements developed all pass a patch test. The resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated. Copyright © 1999 John Wiley & Sons, Ltd.     

An improved three-noded triangular element for plate bending

A new three-noded triangular element for plate bending is described. The element is based on an earlier stress-smoothed triangular element due to Razzaque,¹ but extra internal ‘bubble’ functions are included to make it more flexible. The accuracy of the new element is compared with that of a number of other high-performance triangular elements. It is concluded that the present element and that due to Hansen, Bergan and Syvertsen² are the two most accurate triangular thin plate elements currently available. The extra lines of FORTRAN required to convert Razzaque's shape function subroutine to that for the new element are given, thus making the new element easy to implement in any general-purpose finite element program.