On the use of isoparametric finite elements in linear fracture mechanics

Quadratic isoparametric elements which embody the inverse square root singularity are used in the calculation of stress intensity factors of elastic fracture mechanics. Examples of the plane eight noded isoparametric element show that it has the same singularity as other special crack tip elements, and still includes the constant strain and rigid body motion modes. Application to three-dimensional analysis is also explored. Stress intensity factors are calculated for mechanical and thermal loads for a number of plane strain and three-dimensional problems.     

A comparison between isoparametric Lagrangian elements in 2D BEM

A comparison between a family of isoparametric C°-continucus elements in the context of two-dimensional elastostatics using the direct boundary element method is presented. The elements studied use Lagrangian shape functions of orders quadratic, cubic and quartic. The relative efficiencies of these elements from the viewpoint of accuracy of solution as well as cost effectiveness are examined through certain selected problems. It emerges from the present study that the quartic element is more reliable in accuracy than the lower members of the family without being necessarily more expensive.     

A predictor-corrector contouring algorithm for isoparametric 3D elements

This paper describes a new contouring algorithm for isoparametric elements which can be used to map three-dimensional scalar fields. The contours are generated on arbitrary planes intersecting finite element structures. Tracing element contour lines is accomplished by an accurate predictor-corrector technique. A method of finding starting points for the algorithm on the boundary of the elements is also given.     

An assessment of crack tip singularity models for use with isoparametric elements

The use of finite element methods to analyse fracture problems is complicated by the stress field singularity which exists at the crack tip. The two most successful methods of approach would appear to be the so-called energy technique and the singularity function formulation. The necessity for extremely fine meshes in the crack tip region can be overcome by the use of special elements which incorporate the required stress singularity in their formulation. The aim of this paper is to develop various promising singularity function elements and assess their performance in the solution of standard test problems. These elements are based on the eight node parabolic isoparametric element; this being the most popular element in general use. Such crack tip elements may be readily incorporated into a mesh of standard isoparametric elements permitting numerical fracture studies to be undertaken without extensive mesh regeneration or refinement. In particular elements based on the use of distorted shape functions, standard shape functions, analytic solutions, a superposition process and a hybrid technique are considered. Test problems of both single and combined mode fracture are employed in the assessment of each model.It is also demonstrated that the hybrid element is a special case of the boundary integral method, and suggestions are made for possible future development.     

Control of analyses with isoparametric elements in both 2‐D and 3‐D

This paper presents an extension of the computation method for the error in the constitutive relation to include isoparametric elements. Developed over the past few years at ENS Cachan's laboratory, this technique is based on the strict building of admissible displacement and stress fields. This building process, validated for several types of 2‐D and 3‐D straight finite elements (triangles and quadrilaterals, tetrahedra and hexahedra), cannot be extended to isoparametric elements. For such elements, the method consists of seeking an approximation of the statically admissible stress field by solving a high‐degree finite element problem on each element. This technique, as implemented in our error computation code, which is associated both with a method of computing optimal sizes and with meshers able to respect a size map, allows us to optimize 2‐D and 3‐D meshes. Examples demonstrate the capabilities of the proposed method. Copyright © 1999 John Wiley & Sons, Ltd.     

2D and 3D calculations of micromagnetic wall structures using finite elements

To compute wall structures in a uniaxial magnetic garnet, static micromagnetic equations are solved using the finite-element method. After setting up the physical and the mathematical context, the discretized equations are presented and discretized by standard first-order brick finite elements. A descent method, solving the nonlinear problem, is discussed, and some simulation results are given. This method is found to be time-consuming in 3-D, and a general framework for an adapted mesh algorithm that will speed up the computations is proposed.<>     

On quadratic isoparametric transition elements for a crack normal to a bimaterial interface

The quadratic isoparametric crack‐tip elements proposed by R. E. Abdi and G. Valentin (Computers and Structures 33, 241–248) are reconsidered and a simpler method for calculating the optimal position of the side nodes proposed. Quadratic isoparametric transition elements for an r^λ−1 (0<λ<1) strain singularity are formulated. The effects of these transition elements on the accuracy of the calculated stress intensity factors are shown numerically for a crack normal to and terminating at a bimaterial interface. Finally, layered transition elements are formulated for this case and their effects studied numerically. Copyright © 1999 John Wiley & Sons, Ltd.     

An experimental investigation of the use of isoparametric and rational basis elements in a fluid flow problem

Three competing finite element techniques are applied to a simple flow problem in a region with curved boundaries. Only three-sided elements are considered. The techniques are (1) straight-sided elements, (2) isoparametric elements, both of which use polynomial bases, and (3) curved elements using rational basis functions. The broad conclusions are that both curved-edge techniques are much superior to their straight-sided equivalent and that rational basis functions are quite successful in practice but require slightly more effort to implement, and cost a little more in computer time. From the point of view of efficiency, isoparametric elements seem to be the best choice.     

Mathematical simulation of singularities for isoparametric elements of arbitrary orders

For the general quadrilateral isoparametric elements with 4(n?1) nodes (i.e. n nodes along each side), it is shown that the inverse square root singularity of the strain field at the crack tip can be obtained by a general but simple rule. This amounts to collapsing the quadrilateral elements into triangular elements around the crack tip and placing the (n?2) mid-side nodes at locations γ²/(n?1)² (γ=1,2,…, n?2) times the length of side emanating from the crack tip. These locations are measured from the crack tip. The known results for n = 3 and n = 4 are thus obtained as special cases.     

Stabilization of the zero-energy modes of under-integrated isoparametric finite elements

The coefficient matrices of isoparametrically distorted finite elements are frequently computed by numerical integration formulas with a lower degree of exactness than required by the degree of the integrand. This may result in a rank deficiency of the element matrices which causes an oscillatory instability in the solution. To avoid this effect the element matrices may be stabilised, i. e. their rank may be raised artificially. A new approach is suggested which approximates exact integration without the computational expense usually connected with it. For this purpose, a new family of integration formulas is introduced which is based on the standard Gauss product formulas extended by derivatives of the integrand. For the stabilization, these derivatives can be approximated very easily. The procedure generates a stabilization matrix which, when added to the under-integrated matrix, produces the correct rank. If e. g. shear locking is a problem, the stabilization matrix may be scaled down. Two- and three-dimensional Lagrange elements in second and fourth order problems and Mindlin plate elements are presented together with the results of computations and numerical experiments.     

Effects of element distortions on the performance of isoparametric elements

We discuss the effects of element distortions on the performance of displacement-based isoparametric quadrilateral finite elements. Suitable example problems for both the Lagrangian and the serendipity types of elements are used to show numerically the effects of element distortions. The effects of angular and curved-edge distortions are evaluated analytically in terms of the abilities of the elements to represent exactly various polynomial displacement fields. We then solve a plane stress problem adaptively to demonstrate the effect of element distortions on the accuracy of the stress distribution. The conclusion is that the Lagrangian elements are not affected by angular distortions and are therefore the more reliable elements for general use.     

Finite element analyses of three dimensional fully plastic solutions using quasi-nonsteady algorithm and tetrahedral elements

To estimate entire elastic-plastic behaviors of cracked bodies, fully plastic solutions are utilized with linear elastic solutions in the engineering approach. Some numerical algorithms such as the Selective Reduced Integration/Penalty Function (SRI/PF) method have been developed and utlized to calculate various two-dimensional fully plastic solutions. However, only a few three-dimensional solutions have been obtained because of their numerical instability caused by the interaction among crack-tip singularity, material nonlinearity and incompressibility. This paper describes a new finite element algorithm for three-dimensional fully plastic solutions. The algorithm is basically classified into the mixed formulations. By introducing an artificial viscosity term to the governing equations, static crack problems are converted into quasi-nonsteady ones, which are solved using the fractional step method. The conversion makes the algorithm stable even in the analyses of complex crack geometries though it would need a number of iterations. In the analyses, mixed interpolation tetrahedral elements are also employed from a viewpoint of high quality mesh generation for three-dimensional cracked geometries. Numerical accuracy of the present algorithm is clearly demonstrated through the analyses of the three-dimensional fully plastic solutions of center cracked plates.     

The performance of isoparametric finite elements in stress intensity factor determination

Analysis of a compact compression specimen used for fracture toughness evaluation of cementitious materials is carried out by the finite element method using isoparametric elements. Both triangular and rectangular elements were used with those surrounding the crack tip being of the quarter point type. Solutions were obtained for different mesh subdivisions and convergenece curves for the stress intensity factor were obtained by several methods based on extrapolation and energy techniques. It is found that monotonic convergence was obtained for all cases considered. Employing uniformly graded rectangular element representations converged solutions for the stress intensity factor (assuming a 1 percent convergence criterion) were obtained by the energy methods using a total of 720 degrees of freedom for solving half the structure.Tests on modified 100 mm cubes with symmetrical notches were conducted to determine the fracture toughness. The fracture toughness was calculated from the stress intensity factor and the maximum load obtained from the tests which were conducted in a stiff Instron testing machine. The fracture toughness is found to be independent of the size of the notch.     

Comparison of mixed and isoparametric boundary elements in time domain poroelasticity

A poroelastodynamic Boundary Element (BE) formulation in time domain based on the Convolution Quadrature Method (CQM) is used to model wave propagation phenomena. In the conventional BEM implementation, the same shape functions are applied to all state variables.Motivated by the improvements due to mixed elements in FEM, i.e. the shape function for the pore pressure is chosen one degree lower than for the displacement, such elements have been added to the BEM implementation in both two dimensional (2-d) and three dimensional (3-d) formulations. A study about the influence of the mixed shape functions to the quality of numerical results and the stability of the time-stepping scheme is presented. The mixed elements increase the numerical cost significantly and the results are inconclusive as to whether they improve the CQM based BEM. Therefore, they are only recommended for special cases, in particular the incompressible model in 2-d, where they tend to a significant reduction of the lower stability limit.