T-elements: State of the art and future trends

Summary Trefftz-type elements, or T-elements, are finite elements the internal field of which fulfils the governing differential equations of the problema priori whereas the interelement continuity and the boundary conditions are enforced in an integral weighted residual sense or pointwise. Although the key ideas of such elements can be traced back to Jirousek and Leon in 1977, the T-element approach has received serious attention only for the past ten years. The T-element approach makes it possible to generate highly accurate h- or p-elements exhibiting many important advantages over their more conventional counterpart. The paper surveys existing T-element formulations (including some yet unpublished ones) and assesses critically their performance (accuracy, h- and p-convergence, sensitivity to mesh distortions, handling of singularities and geometry or load induced local effects, etc.). The available applications include plane elasticity, thin and thick plates, cylindrical shells, axisymmetric 3-D elasticity, Poisson's equation and transient heat conduction analysis. Existing approaches to adaptive reliability assurance based on p-extension are also discussed and future trends in the T-element research shortly outlined. Research engineer, on leave from the Institute of Mechanics and Machine Design     

Dynamic inelastic structural analysis by boundary element methods

Summary Boundary element methodologies for the determination of the response of inelastic two-and three-dimensional solids and structures as well as beams and flexural plates to dynamic loads are briefly presented and critically discussed. Elastoplastic and viscoplastic material behaviour in the framework of small deformation theories are considered. These methodologies can be separated into four main categories: those which employ the elastodynamic fundamental solution in their formulation, those which employ the elastostatic fundamental solution in their formulation, those which combine boundary and finite elements for the creation of an efficient hybrid scheme and those representing special boundary element techniques. The first category, in addition to the boundary discretization, requires a discretization of those parts of the interior domain expected to become inelastic, while the second category a discretization of the whole interior domain, unless the inertial domain integrals are transformed by the dual reciprocity technique into boundary ones, in which case only the inelastic parts of the domain have to be discretized. The third category employs finite elements for one part of the structure and boundary elements for its remaining part in an effort to combine the advantages of both methods. Finally, the fourth category includes special boundary element techniques for inelastic beams and plates and symmetric boundary element formulations. The discretized equations of motion in all the above methodologies are solved by efficient step-by-step time integration algorithms. Numerical examples involving two-and three-dimensional solids and structures and flexural plates are presented to illustrate all these methodologies and demonstrate their advantages. Finally, directions for future research in the area are suggested.     

Large finite elements method for the solution of problems in the theory of elasticity

In contrast to conventional finite element (CFE) formulations, the large finite element (LFE) concept is based on subdividing the region under consideration into a small number of LFE and using in each of them an appropriate parametric displacement field such that the governing differential problem equations are satisfied a priori (Trefftz's method). Where relevant, known local solutions in the vicinity of a stress concentration or stress singularity are used as a convenient expansion basis. The boundary conditions, as well as the continuity across the interfaces, are implicitly imposed by an appropriate variational functional.

The LFE concept attempts to combine the flexibility of the conventional FE method with the accuracy and high convergence rate of the Trefftz's method. The paper summarises the principal results obtained and shows that the practical efficiency of the LFE analysis is superior to a CFE solution, for both regular and singular problems.     

A coupled formulation of finite and boundary element methods in two-dimensional elasticity

A symmetric stiffness formulation based on a boundary element method is studied for the structural analysis of a shear wall, with or without cutouts. To satisfy compatibility requirements with finite beam elements and to avoid problems due to the eventual discontinuities of the traction vector, different interpolation schemes are adopted to approximate the boundary displacements and tractions. A set of boundary integral equations is obtained with the collocation points on the boundary, which are selected by the error minimization technique proposed in this paper, and the stiffness matrix is formulated with those equations and symmetric coupling techniques of finite and boundary element methods. The newly developed plane stress element can have the openings in its interior domain and can be easily linked with the finite beam/column elements.     

A new finite element formulation for computational fluid dynamics: VII. The stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces

Symmetric finite element formulations are proposed for the primitive-variables form of the Stokes equations and shown to be convergent for any combination of pressure and velocity interpolations. Various boundary conditions, such as pressure, are accommodated.     

Parallel Semi-Implicit Spectral Element Methods for Atmospheric General Circulation Models

Spectral elements combine the accuracy and exponential convergence of conventional spectral methods with the geometric flexibility of finite elements. Additionally, there are several apparent computational advantages to using spectral element methods on microprocessors. In particular, the computations are naturally cache-blocked and derivatives may be computed using nearest neighbor communications. Thus, an explicit spectral element atmospheric model has demonstrated close to linear scaling on a variety of distributed memory computers including the IBM SP and Linux Clusters. Explicit formulations of PDE's arising in geophysical fluid dynamics, such as the primitive equations on the sphere, are time-step limited by the phase speed of gravity waves. Semi-implicit time integration schemes remove the stability restriction but require the solution of an elliptic BVP. By employing a weak formulation of the governing equations, it is possible to obtain a symmetric Helmholtz operator that permits the solution of the implicit problem using conjugate gradients. We find that a block-Jacobi preconditioned conjugate gradient solver accelerates the simulation rate of the semi-implicit relative to the explicit formulation for practical climate resolutions by about a factor of three.     

p-Version axisymmetric shell element for geometrically nonlinear analysis

This paper presents a p-version geometrically nonlinear (GNL) formulation based on total Lagrangian approach for a three-node axisymmetric curved shell element. The approximation functions and the nodal variables for the element are derived directly from the Lagrange family of interpolation functions of order p_ξ and p_η. This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and η directions for the three- and one-node equivalent configurations that correspond to p_ξ + 1 and p_η+ 1 equally spaced nodes in the ξ and η directions and then taking their products. The resulting element approximation functions and the nodal variables are hierarchical and the element approximation ensures C⁰ continuity. The element geometry is described by the coordinates of the nodes located on the middle surface of the element and the nodal vectors describing top and bottom surfaces of the element.

The element properties are established using the principle of virtual work and the hierarchical element approximation. In formulating the properties of the element complete axisymmetric state of stresses and strains are considered, hence the element is equally effective for very thin as well as extremely thick shells. The formulation presented here removes virtually all of the drawbacks present in the existing GNL axisymmetric shell finite element formulations and has many additional benefits. First, the currently available GNL axisymmetric shell finite element formulations are based on fixed interpolation order and thus are not hierarchical and have no mechanism for p-level change. Secondly, the element displacement approximations in the existing formulations are either based on linearized (with respect to nodal rotation) displacement fields in which case a true Lagrangian formulation is not possible and the load step size is severely limited or are based on nonlinear nodal rotation functions approach in which case though the kinematics of deformation is exact but additional complications arise due to the noncummutative nature of nonlinear nodal rotation functions. Such limitations and difficulties do not exist in the present formulation. The hierarchical displacement approximation used here does not involve traditional nodal rotations that have been used in the existing shell element formulations, thus the difficulties associated with their use are not present in this formulation.

Incremental equations of equilibrium are derived and solved using the standard Newton method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the preset formulation. The results obtained from the present formulation are compared with those available in the literature.     

Singularly perturbed reaction–diffusion equations in a circle with numerical applications

The goal of this article is to study the boundary layers of reaction–diffusion equations in a circle and provide some numerical applications which utilize the so-called boundary layer elements. Via the boundary layer analysis, we obtain the valid asymptotic expansions at any order and devise boundary layer elements to be conveniently used in the finite element schemes. Using boundary layer elements incorporated in the finite element space, we obtain accurate numerical solutions in a quasi-uniform mesh with convergence of order 2.     

Finite element method in plane Cosserat elasticity

A displacement and rotation based finite element method for the solution of boundary value problems in linear isotropic Cosserat elasticity is proposed. The field equations for the problem of plane strain are derived from the three-dimensional theory and expressed in oblique rectilinear coordinates. Three fundamental triangular finite elements are presented. A patch test for validating Cosserat finite element formulations is also introduced.     

A hybrid finite element-scaled boundary finite element method for crack propagation modelling

This study develops a novel hybrid method that combines the finite element method (FEM) and the scaled boundary finite element method (SBFEM) for crack propagation modelling in brittle and quasi-brittle materials. A very simple yet flexible local remeshing procedure, solely based on the FE mesh, is used to accommodate crack propagation. The crack-tip FE mesh is then replaced by a SBFEM rosette. This enables direct extraction of accurate stress intensity factors (SIFs) from the semi-analytical displacement or stress solutions of the SBFEM, which are then used to evaluate the crack propagation criterion. The fracture process zones are modelled using nonlinear cohesive interface elements that are automatically inserted into the FE mesh as the cracks propagate. Both the FEM’s flexibility in remeshing multiple cracks and the SBFEM’s high accuracy in calculating SIFs are exploited. The efficiency of the hybrid method in calculating SIFs is first demonstrated in two problems with stationary cracks. Nonlinear cohesive crack propagation in three notched concrete beams is then modelled. The results compare well with experimental and numerical results available in the literature.     

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.     

Higher order meshless schemes applied to the finite element method in elliptic problems

This paper presents selected approximation techniques, typical for the meshless finite difference method (MFDM), although applied to the finite element method (FEM). Finite elements with standard or hierarchical shape functions are coupled with higher order meshless schemes, based upon the correction terms of a simple difference operator. Those terms consist of higher order derivatives, which are evaluated by means of the appropriate formulas composition as well as a numerical solution, which corresponds to the primary interpolation order, assigned to element shape functions. Correction terms modify the right-hand sides of algebraic FE equations only, yielding an iterative procedure. Therefore, neither re-generation of the stiffness matrix nor introduction of any additional nodes and/or degrees of freedom is required. Such improved FE-MFD solution approach allows for the optimal application of advantages of both methods, for instance, a high accuracy of the nodal FE solution and a derivatives’ super-convergence phenomenon at arbitrary domain points, typical for the meshless FDM. Existing and proposed higher order techniques, applied in the FEM, are compared with each other in terms of the solution accuracy, algorithm efficiency and computational complexity.In order to examine the considered algorithms, numerical results of several two-dimensional benchmark elliptic problems are presented. Both the accuracy of a solution and the solution’s derivatives as well as their convergence rates, evaluated on irregular and structured meshes as well as arbitrarily irregular adaptive clouds of nodes, are taken into account.     

Temporal finite element formulation of optimal control in mechanisms

A temporal finite element discretization of a boundary value problem has several advantages compared to a time-integrating evolution form for optimized target movement simulations. The paper gives some basic aspects on how such a finite element form can be stated, with both displacements and controls discretized and seen as unknowns. Aspects on the resulting formulations are discussed. Important issues are the order, continuity and fineness of the discretizations. When the formulation is seen in an optimization context, minimizing the effort for a prescribed movement, the discretization affects the results obtained in several manners, where some aspects of results are artifacts. The paper discusses these effects from basic principles, but also verifies them in numerical simulations.     

A new method for the coupling of finite element and boundary element discretized subdomains of elastic bodies

A new and efficient approach for the coupling of subregions of elastic solids discretized by means of finite elements (FE) and boundary elements (BE), respectively, is presented. The method is characterized by so-called ‘bi-condensation’ of nodal degrees of freedom followed by the transformation of the resulting BEM-related traction-displacement equations for the interface(s) of the BE subregion(s) and the FE subdomain(s) to ‘FEM-like’ force-displacement relations which are assembled with the FEM-related force-displacement equations for the interface(s). The presented ‘local FE coupling approach’ is computationally more economic than a global coupling approach since it only requires the inversion of BEM-related coefficient matrices referred to the interfaces of BE subregions and FE subdomains. Depending on whether the principle of virtual displacements or the principle of minimum of potential energy is used for the generation of force-displacement equations for the coupling interface(s), unsymmetric or symmetric coefficient matrices are obtained. Since the two principles are mechanically equivalent, identical results would be achieved in the limit of finite discretizations.The numerical investigation has shown that, depending on the problem and the discretization, the results obtained on the basis of symmetric coefficient matrices may be poor. This applies to ‘edge problems’ characterized by discontinuous tractions along the edges. On the basis of unsymmetric coefficient matrices, however, satisfactory results are obtained even for relatively coarse discretizations.     

Asymptotic and finite element approximations for heat transfer in rotating compressible flow over an infinite porous disk

The laminar boundary layer equations for the compressible flow due to the finite difference in rotation and temperature rates are solved for the case of uniform suction through the disk. The effects of viscous dissipation on the incompressible flow are taken into account for any rotation rate, whereas for a compressible fluid they are considered only for a disk rotating in a stationary fluid. For the general case, the governing equations are solved numerically using a standard finite element scheme. Series solutions are developed for those cases where the suction effect is dominant. Based on the above analytical and numerical solutions, a new asymptotic finite element scheme is presented. By using this scheme one can significantly improve the pointwise accuracy of the standard finite element scheme.     

Mixed finite element models for plate bending analysis theory

The theoretical background of mixed finite element models, in general for nonlinear problems, is briefly reexamined. In the first part of the paper, several alternative “mixed” formulations for 3-D continua undergoing large elastic deformations under the action of time dependent external loading are outlined and are examined incisively. It is concluded that mixed finite element formulations, wherein the interpolants for the stress field satisfy only a part of the domain equilibrium equations, are not only consistent from a theoretical standpoint but are also preferable from an implementation point of view. In the second part of the paper, alternative variational bases for the development of thin-plate elements are presented and discussed in detail. In light of this discussion, it is concluded that the “bad press” generated in the past concerning the practical relevance of the so-called assumed stress hybrid finite element model is not justified. Moreover, the advantages of this type of elements as compared with the “assumed displacement” or alternative mixed elements are outlined.     

A comparison of finite-difference and finite-element methods for calculating free edge stresses in composites

This study considers the accuracy of the finite difference method in the solution of linear elasticity problems that involve either a stress discontinuity or a stress singularity. Solutions to three elasticity problems are discussed in detail: a semi-infinite plane subjected to a uniform load over a portion of its boundary; a bimetallic plate under uniform tensile stress; and a long, midplane symmetric, fiber-reinforced laminate subjected to uniform axial strain. Finite difference solutions to the three problems are compared with finite element solutions to corresponding problems. For the first problem a comparison with the exact solution is also made. The finite difference formulations for the three problems are based on second order finite difference formulas that provide for variable spacings in two perpendicular directions. Forward and backward difference formulas are used near boundaries where their use eliminates the need for fictitious grid points. Moreover, forward and backward finite difference formulas are used to enforced continuity of interlaminar stress components for the third problem. The study shows that the finite difference method employed in this investigation provides solutions to the three elasticity problems considered that are as accurate as the corresponding finite element solutions. Furthermore, the finite difference method appears to give a solution for the laminate problem that characterizes the stress distributions near an interface corner in a more realistic manner than the finite element method.