p-convergent finite element approximations in fracture mechanics

The strain energy release rate (G) converges rapidly in finite element approximations in which the finite element mesh is fixed and the order of polynomial displacement interpolations (p) is increased. Numerical experiments indicate that the error inG is very closely estimated, even for small pand very coarse finite element meshes, by an expression of the form k (NDF)^-1 in which k is a mesh dependent constant and NDF is the number of degrees-of-freedom. The method provides for very efficient and accurate computation of G without the use of special techniques.     

Automatic element reordering for finite element analysis with frontal solution schemes

This paper describes an element reordering algorithm which is suitable for use with a frontal solution package. The procedure is shown to generate efficient element numberings for a wide variety of test examples. In an effort to obtain an optimum elimination order, the algorithm first renumbers the nodes, and then uses this result to resequence the elements. This intermediate step is necessary because of the nature of the frontal solution procedure, which assembles variables on an element-by-element basis but eliminates them node by node. To renumber the nodes, a modified version of the King¹ algorithm is used. In order to minimize the number of nodal numbering schemes that need to be considered, the starting nodes are selected automatically by using some concepts from graph theory. Once the optimum numbering sequence has been ascertained, the elements are then reordered in an ascending sequence of their lowest-numbered nodes. This ensures that the new elimination order is preserved as closely as possible. For meshes that are composed of a single type of high-order element, it is only necessary to consider the vertex nodes in the renumbering process. This follows from the fact that mesh numberings which are optimal for low-order elements are also optimal for high-order elements. Significant economies in the reordering strategy may thus be achieved. A computer implementation of the algorithm, written in FORTRAN IV, is given.     

Elastic-plastic finite element analysis of thermally loaded cracked structures

Two elastic-plastic fracture parameters are considered here,J* and a direct evaluation of the potential energy release rate,G. Both parameters are evaluated from elastic-plastic finite element solutions of thermally loaded structures. The results show a reasonable agreement between the two parameters withG being numerically more stable.J* can be sensitive to mesh details and loading.

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Résumé On considère dans ce mémoire deux paramètres gouvernant une rupture élastoplastique, l'intégraleJ* et une évaluation directeG de la vitesse de relaxation de l'énergie potentielle. Ces deux paramètres sont évalués à partir d'éléments finis élasto-plastiques, dans le cas de composants sollicités thermiquement. Les résultats font état d'un accord satisfaisant entre les deux paramètres, bien queG apparaissent numériquement plus stable. Le paramètreJ* peut être sensible à des détails du maillage et à sa mise en charge.

     

The quasi-uniformity condition for reproducing kernel element method meshes

The reproducing kernel element method is a hybrid between finite elements and meshfree methods that provides shape functions of arbitrary order and continuity yet retains the Kronecker-δ property. To achieve these properties, the underlying mesh must meet certain regularity constraints. This paper develops a precise definition of these constraints, and a general algorithm for assessing a mesh is developed. The algorithm is demonstrated on several mesh types. Finally, a guide to generation of quasi-uniform meshes is discussed.     

An alternative pseudoperipheral node finder for resequencing schemes

Many resequencing algorithms for reducing the bandwidth, profile and wavefront of sparse symmetric matrices have been published. In finite element applications, the sparsity of a matrix is related to the nodal ordering of the finite element mesh. Some of the most successful algorithms, which are based on graph theory, require a pair of starting pseudoperipheral nodes. These nodes, located at nearly maximal distance apart, are determined using heuristic schemes. This paper presents an alternative pseadoperipheral node finder, which is based on the algorithm developed by Gibbs, Poole and Stockmeyer. This modified scheme is suitable for nodal reordering of finite meshes and provides more consistency in the effective selection of the starting nodes in problems where the selection becomes arbitrary due to the number of candidates for these starting nodes. This case arises, in particular, for square meshes. The modified scheme was implemented in Gibbs-Poole-Stockmeyer, Gibbs-King and Sloan algorithms. Test problems of these modified algorithms include: (1) Everstine's 30 benchmark problems; (2) sets of square, rectangular and annular (cylindrical) finite element meshes with quadrilateral and triangular elements; and (3) additional examples originating from mesh refinement schemes. The results demonstrate that the modifications to the original algorithms contribute to the improvement of the reliability of all the resequencing algorithms tested herein for the nodal reordering of finite element meshes.     

An efficient finite element solution using a large pre-solved regular element

In this paper a finite element algorithm is presented using a large pre-solved hyper element. Utilizing the largest rectangle/cuboid inside an arbitrary domain, a large hyper element is developed that is solved using graph product rules. This pre-solved hyper element is efficiently inserted into the finite element formulation of partial differential equations (PDE) and engineering problems to reduce the computational complexity and execution time of the solution. A general solution of the large pre-solved element for a uniform mesh of triangular and rectangular elements is formulated for second-order PDEs. The efficiency of the algorithm depends on the relative size of the large element and the domain; however, the method remains as efficient as a classic method for even relatively small sizes. The application of the method is demonstrated using different examples.     

Numerical evaluation of the quarter-point crack tip element

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.     

Mesh refinement and iterative solution methods for finite element computations

Global and element residuals are introduced to determine a posteriori, computable, error bounds for finite element computations on a given mesh. The element residuals provide a criterion for determining where a finite element mesh requires refinement. This indicator is implemented in an algorithm in a finite element research program. There it is utilized to automatically refine the mesh for sample two-point problems exhibiting boundary layer and interior layer solutions. Results for both linear and nonlinear problems are presented. An important aspect of this investigation concerns the use of adaptive refinement in conjunction with iterative methods for system solution. As the mesh is being enriched through the refinement process, the solution on a given mesh provides an accurate starting iterate for the next mesh, and so on. A wide range of iterative methods are examined in a feasibility study and strategies for interweaving refinement and iteration are compared.     

An algorithm for adaptive refinement of triangular element meshes

A simple algorithm is developed for adaptive and automatic h refinement of two-dimensional triangular finite element meshes. The algorithm is based on an element refinement ratio that can be calculated from an a posteriori error indicator. The element subdivision algorithm is robust and recursive. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. Several example problems are presented to illustrate the utility of the approach.     

Adaptive solution strategy for solving large systems of p-type finite element equations

A solution strategy is proposed and implemented for taking advantage of the hierarchical structure of linear equation sets arising from the p-type finite element method using a hierarchical basis function set. The algorithm dynamically branches to either direct or iterative solution methods. In. the iterative solution branch, the substructure of the finite element equation set is used to generate a lower order preconditioner for a preconditioned conjugate gradient (PCG) method. The convergence rate of the PCG algorithm is monitored to improve the heuristics used in the choice of the preconditioner. The robustness and efficiency of the method are demonstrated on a variety of three dimensional examples utilizing both hexahedral and tetrahedral mesh discretizations. This strategy has been implemented in a p-version finite element code which has been used in an industrial environment for over two years to solve mechanical design problems.     

An adaptive hp-version of the finite element method applied to flame propagation problems

This paper describes an adaptive hp-version mesh refinement strategy and its application to the finite element solution of one-dimensional flame propagation problems. The aim is to control the spatial and time discretization errors below a prescribed error tolerance at all time levels. In the algorithm, the optimal time step is first determined in an adaptive manner by considering the variation of the computable error in the reaction zone. Later, the method uses a p-version refinement till the computable a posteriori error is brought down below the tolerance. During the p-version, if the maximum allowable degree of approximation is reached in some elements of the mesh without satisfying the global error tolerance criterion, then conversion from p- to h-version is performed. In the conversion procedure, a gradient based non-uniform h-version refinement has been introduced in the elements of higher degree approximation. In this way, p-version and h-version approaches are used alternately till the a posteriori error criteria are satisfied. The mesh refinement is based on the element error indicators, according to a statistical error equi-distribution procedure. Numerical simulations have been carried out for a linear parabolic problem and premixed flame propagation in one-space dimension. © 1997 John Wiley & Sons, Ltd.     

Automatic triangulation of arbitrary planar domains for the finite element method

The object of this paper is to describe a new algorithm for the semi-automatic triangulation of arbitrary, multiply connected planar domains. The strategy is based upon a modification of a finite element mesh genration algorithm recently developed. ¹ The scheme is designed for maximum flexibility and is capable of generating meshes of triangular elements for the decomposition of virtually any multiply connected planar domain. Moreover, the desired density of elements in various regions of the problem domain is specified by the user, thus allowing him to obtain a mesh decomposition appropriate to the physical loading and/or boundary conditions of the particular problem at hand. Several examples are presented to illustrate the applicability of the algorithm. An extension of the algorithm to the triangulation of shell structures is indicated.     

Nearly Completely Positive Graphs

 A non-cp graph G is called nearly vertex (edge) cp if any subgraph of G obtained by deleted a vertex (an edge) of G is completely positive. This paper presents necessary and sufficient conditions for any doubly nonnegative matrix realization A of a nearly cp graph to be completely positive. Received: December 14, 1999; revised version: October 5, 2000     

A hierarchical approach to automatic finite element mesh generation

A point-based two-stage hierarchical method for automatic finite element mesh generation from a solid model is presented. Given the solid model of a component and the required nodal density distribution, nodes are generated according to the hierarchy—vertex, edge, face and solid. At the vertices, nodes are established naturally. Nodes on the edges, faces and inside the solid model are generated by recursive subdivision. The nodes are then connected to form a valid and well-conditioned finite element mesh of tetrahedron elements using modified Delaunay Triangulation. Checks are conducted to ensure the compatibility of geometry and topology between the solid model and the mesh.     

Refined h-adaptive finite element procedure for large deformation geotechnical problems

Adaptive finite element procedures automatically refine, coarsen, or relocate elements in a finite element mesh to obtain a solution with a specified accuracy. Although a significant amount of research has been devoted to adaptive finite element analysis, this method has not been widely applied to nonlinear geotechnical problems due to their complexity. In this paper, the h-adaptive finite element technique is employed to solve some complex geotechnical problems involving material nonlinearity and large deformations. The key components of h-adaptivity including robust mesh generation algorithms, error estimators and remapping procedures are discussed. This paper includes a brief literature review as well as formulation and implementation details of the h-adaptive technique. Finally, the method is used to solve some classical geotechnical problems and results are provided to illustrate the performance of the method.     

An adaptive non‐conforming finite‐element method for Reissner–Mindlin plates

Adaptive algorithms are important tools for efficient finite‐element mesh design. In this paper, an error controlled adaptive mesh‐refining algorithm is proposed for a non‐conforming low‐order finite‐element method for the Reissner–Mindlin plate model. The algorithm is controlled by a reliable and efficient residual‐based a posteriori error estimate, which is robust with respect to the plate's thickness. Numerical evidence for this and the efficiency of the new algorithm is provided in the sense that non‐optimal convergence rates are optimally improved in our numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

An adaptive remeshing procedure for discontinuous finite element limit analysis

An adaptive remeshing procedure is proposed for discontinuous finite element limit analysis. The procedure proceeds by iteratively adjusting the element sizes in the mesh to distribute local errors uniformly over the domain. To facilitate the redefinition of element sizes in the new mesh, the interelements discontinuous field of elemental bound gaps is converted into a continuous field, ie, the intensity of bound gap, using a patch‐based approximation technique. An analogous technique is subsequently used for the approximation of element sizes in the old mesh. With these information, an optimized distribution of element sizes in the new mesh is defined and then scaled to match the total number of elements specified for each iteration in the adaptive remeshing process. Finally, a new mesh is generated using the advancing front technique. This adaptive remeshing procedure is repeated several times until an optimal mesh is found. Additionally, for problems involving discontinuous boundary loads, a novel algorithm for the generation of fan‐type meshes around singular points is proposed explicitly and incorporated into the main adaptive remeshing procedure. To demonstrate the feasibility of our proposed method, some classical examples extracted from the existing literary works are studied in detail.     

A unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations

Piecewise linear finite element approximations to two-dimensional Poisson problems are treated. For simplicity, consideration is restricted to problems having Dirichlet boundary conditions and defined on rectangular domains Ω which are partitioned by a uniform triangular mesh. It is also required that the solutions u ∈ H³ (Ω). A method is proposed for recovering the gradients of the finite element approximations to a root mean square accuracy of O(h²), both at element edge mid-points and element vertices, using simple averaging schemes over adjacent elements. Piecewise linear interpolants (respectively discontinuous and continuous) are then fitted to these recovered gradients, and are shown to be O(h²) estimates for ?u in the L₂-norm, and thus superconvergent. A discussion is given of the extension of the results to problems with more general region and mesh geometries, boundary conditions and with solutions of lower regularity, and also to other second-order elliptic boundary value problems, e.g. the problem of planar linear elasticity.     

C0 finite element discretization of Kirchhoff's equations of thin plate bending

An alternative formulation of Kirchhoff's equations is given which is amenable to a standard C⁰ finite element discretization. In this formulation, the potential energy of the plate is formulated entirely in terms of rotations, whereas the deflections are the outcome of a subsidiary problem. The nature of the resulting equations is such that C⁰ interpolation can be used on both rotations and deflections. In particular, general classes of triangular and quadrilateral isoparametric elements can be used in conjunction with the method. Unlike other finite element methods which are based on three-dimensional or Mindlin formulations, the present approach deals directly with Kirchhoff's equations of thin plate bending. Excellent accuracy is observed in standard numerical tests using both distorted and undistorted mesh patterns.