No abstract

In the p-version of the finite element method, the triangulation is fixed and the degree p, of the piecewise polynomial approximation, is progressively increased until some desired level of precision is reached.In this paper, we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation. These properties lead to an a priori estimate of the asymptotic rate of convergence of the p-version. The estimate shows that the p-version gives results which are not worse than those obtained by the conventional finite element method (called the h-version, in which h represents the maximum diameter of the elements), when quasi-uniform triangulations are employed and the basis for comparison is the number of degrees of freedom. Furthermore, in the case of a singularity problem, we show (under conditions which are usually satisfied in practice) that the rate of convergence of the p-version is twice that of the h-version with quasi-uniform mesh. Inverse approximation theorems which determine the smoothness of a function based on the rate at which it is approximated by piecewise polynomials over a fixed triangulation are proved for both singular and nonsingular problems.We present numerical examples which illustrate the effectiveness of the p-version for a simple one-dimensional problem and for two problems in two-dimensional elasticity. We also discuss roundott error and computational costs associated with the p-version. Finally, we describe some important features, such as hierarchic basis functions, which have been utilized in COMET-X, an experimental computer implementation of the p-version.

No abstract

The rate of convergence of the finite element method is a function of the strategy by which the number of degrees of freedom are increased.Alternative strategies are examined in the light of recent theoretical results and computational experience.

The theoretical basis and performance characteristics of two new methods for the computation of the coefficients of the terms of asymptotic expansions at reentrant corners from finite-element solutions are presented. The methods, called the contour integral method (CIM) and the cutoff function method (CFM), are very efficient: the coefficients converge to their true values as fast as the strain energy, or faster. In order to make the presentation as simple as possible, we assume that the elastic body is homogeneous and isotropic, is loaded by boundary tractions only, and, in the neighborhood of the reentrant corner, has stress-free boundaries. The methods described herein can be adapted to cases without such restrictions.

A method is developed and validated for approximating continuous smooth distributions of finite strains in the ventricles from the deformations of magnetic resonance imaging (MRI) tissue tagging "tag lines" or "tag surfaces." Tag lines and intersections of orthogonal tag lines are determined using a semiautomated algorithm. Three-dimensional (3-D) reconstruction of the displacement field on tag surfaces is performed using two orthogonal sets of MRI images and employing spline surface interpolation. The 3-D regional ventricular wall strains are computed from an initial reference image to a deformed image in diastole or systole by defining a mapping or transformation of space between the two states. The resultant mapping is termed the measurement analysis solution and is defined by determining a set of coefficients for the approximating functions that best fit the measured tag surface displacements. Validation of the method is performed by simulating tag line or surface deformations with a finite element (FE) elasticity solution of the heart and incorporating the measured root-mean-square (rms) errors of tag line detection into the simulations. The FE-computed strains are compared with strains calculated by the proposed procedure. The average difference between two-dimensional (2-D) FE-computed strains and strains calculated by the measurement analysis was 0.022 +/- 0.009 or 14.2 +/- 3.6% of the average FE elasticity strain solution. The 3-D displacement reconstruction errors averaged 0.087 +/- 0.002 mm or 2.4 +/- 0.1% of the average FE solution, and 3-D strain fitting errors averaged 0.024 +/- 0.011 or 15.9 +/- 2.8% of the average 3-D FE elasticity solution. When the rms errors in tag line detection were included in the 2-D simulations, the agreement between FE solution and fitted solution was 24.7% for the 2-D simulations and 19.2% for the 3-D simulations. We conclude that the 3-D displacements of MRI tag lines may be reconstructed accurately; however, the strain solution magnifies the small errors in locating tag lines and reconstructing 3-D displacements.

In the first part of this chapter, the basic algorithmic structure and performance characteristics of the p ‐version of the finite element method (FEM) are surveyed with reference to elliptic problems in solid mechanics. For this class of problems, the theoretical basis of the p ‐version is fully established, and a very substantial amount of engineering experience covering linear and nonlinear applications is available. It is shown that p ‐extensions on properly designed meshes make realization of exponential rates of convergence in practical computations possible and provide for the estimation and control of relative errors in terms of any quantity of interest. In the second part, the p ‐version of the FEM is extended to a high‐order fictitious domain approach, the finite cell method (FCM). While the FCM inherits the advantages of the p ‐version with respect to accuracy and robustness, it relieves analysts from the necessity of generating finite element meshes. Thus, it strongly supports the analysis of problems with highly complicated geometry, for which meshing with finite elements would be very difficult. Given the growing demand for verified numerical solutions, the p ‐version of the finite element and FCMs are expected to play an increasingly important role.

A numerical method for the computation of the generalized flux/stress intensity factors (GFIFs/GSIFs) for the asymptotic solution of linear second-order elliptic partial differential equations in two dimensions in the vicinity of singular points is described. Special attention is given to heat transfer and elasticity problems. The singularities may be caused by re-entrant corners and abrupt changes in material properties.Such singularities are of great interest from the point of view of failure initiation: The eigenpairs, computed in a companion paper,' characterize the straining modes and their amplitudes (the GFIFs/GSIFs) quantify the amount of energy residing in particular straining modes. For this reason, failure theories directly or indirectly involve the GFIFs/GSIFs. This paper addresses a general method based on the complementary weak formulation for determining the GFIFs/GSIFs numerically as a post-solution operation on the finite element solution vector. Importantly, the method is applicable to anisotropic materials, multi-material interfaces, and cases where the singularities are characterized by complex eigenpairs. An error analysis is sketched and numerical examples are presented to illustrate the effectiveness of the technique.