Ling Fu Zeng

Spatial mesh adaptation in semidiscrete finite element analysis of linear elastodynamic problems

A spatial mesh adaptation procedure in semidiscrete finite element analysis of 2D linear elastodynamic problems is presented. The procedure updates, through an automatic remeshing scheme, the spatial mesh when found necessary in order to gain control of the spatial discretization error from time to time. An a posteriori error estimate developed by Zienkiewicz and Zhu (1987) for elliptic problems is extended to dynamic analysis to estimate the spatial discretization error at a certain time, which is found to be reasonable by analyzing an a priori error estimate. Numerical examples are used to demonstrate the performance of the procedure. It is indicated that the extended error estimation and the procedure are capable of monitoring the moving of steep stress regions by updating the spatial mesh according to a prescribed error tolerance, thus providing a reliable finite element solution in an efficient manner.

Error estimation and adaptivity in elastodynamics

Abstract The paper discusses adaptive procedures for elastodynamic problems using semi-discrete finite elements. The adaptivity in space may be done by mesh refinement (h-version), by increase of order of the approximation polynomials (p-version) or a combination of these (hp-version). In time, the integration is either made by mode superposition of characteristic functions or by direct integration. The adaptation in time may either be made by change of the global time-step (h-version) or increase of the order of approximation polynomials (p-version). Crucial parts of the analysis are the error estimator and the error indicator, which are based either on interpolation theory or on total energy obtained from postprocessed stresses. For the mesh generation and regeneration, an isoline technique is utilized by which the mesh can be smoothed by interpolation theory. The isolines are created for some characteristic scalar function. For time discretization, a new simple a posteriori local error estimator for the Newmark scheme is described. It is derived by a post-processing technique without solving new equations so the additional computational cost is small and the implementation is convenient. The paper also describes a sequence of nested time-integration methods for dynamic problems formulated as two first order equations in time. These nested integration methods in time based on a hierarchical formulation in time are A-stable schemes of order 2k and L-stable schemes of order 2k − 1. A considerable improvement in terms of accuracy as well as effectiveness is obtained, compared to currently available methods.

Adaptive h-p procedures for high accuracy finite element analysis of two-dimensional linear elastic problems

Abstract Two adaptive procedures of the h - p version for a high accuracy finite element analysis of two-dimensional elastic problems are studied. These are based on a strategy of first using an h -version to predict a nearly optimal mesh up to a certain accuracy and then following up with a p -version to achieve a higher accuracy. The h -version, using linear triangular elements, is developed by coupling a code, ADMESH, with an error estimation in energy norm. Following the h -version, two alternative procedures of non-uniform p -refinements are then performed. In procedure I, p -refinements are made in one step by selectively adding hierarchical shape functions of order p = 2 and 3 based on the estimated error in energy norm. In procedure II, p -refinements are made in a step-by-step way by which, in the k th step of the p -refinements, hierarchical shape functions of order p = k + 1 are selectively introduced. In the first step of p -refinements of procedure II, hierarchical variables are selected by means of the estimated errors in energy norm, whereas in the later steps, they are selected with a guidance of an error estimate which evaluates the local average error of stresses. The performances of both procedures and the rate of convergence are studied in numerical examples. Numerical tests for the error estimates being used are also made. Obtained results indicate that both procedures can achieve a high accuracy (say, error below 5% measured in energy norm) in an exponential rate of convergence.

Free vibration analysis of Reissner-Mindlin plates using a linked interpolated mixed element

In this paper, a mixed finite element formulation based on Reissner-Mindlin plate theory is applied for the free vibration analysis of linear elastic plates. A four-node quadrilateral element, Q4BL, recently developed by Zienkiewicz et al. [Linked interpolation for Reissner-Mindlin plate elements Part I. Int. J. Numer. Methods Engng (36, 3043–3056 (1993).] using linked interpolation to couple transverse displacement and rotation degrees of freedom, is numerically tested. Numerical examples indicate that the application of such a mixed formulation to dynamic problems is convenient and the element Q4BL is efficient and reliable for both thick and thin plates.

A generalized coordinate method for the analysis of 3D tall buildings

Abstract A generalized coordinate method (GCM) is proposed for the reduction of unknowns in the 3D analysis of tall buildings when the displacement method is employed. The number of variables is reduced by the assumption of in-plane rigid floors and by use of a 2D polynomial approximation for the out-of-plane displacements of floors, and a 1D polynomial approximation of the displacements with the height of building. The overall stiffness equation is obtained in generalized coordinates. The transformations are performed at the member level so that calculations involving large matrices are avoided. The GCM might be considered as a reduction technique based on a combination of the FEM at member level and the Rayleigh-Ritz method at the structure level. The GCM has the advantages that (1) the number of unknowns is significantly reduced and is independent of the number of storeys: (2) the accuracy can be adjusted by selecting the number of terms of the displacement function. The storage needed, the required number of operations and the possibility of choosing hierarchical displacement functions to make the calculation adaptive are discussed. The method presented can easily be extended to nonlinear and dynamic analysis. However, the derivation in this paper is confined only to the linear elastic analysis of frame and frame-shear wall structures. Numerical examples are given to show the efficiency of the proposed method.