Nils-Erik Wiberg

Implementation and Adaptivity of a Space-Time Finite Element Method for Structural Dynamics

Abstract In this paper we study the implementation and adaptivity of a space-time finite element method for the analysis of two-dimensional problems in structural dynamics. The method, also known as the time-discontinuous Galerkin method or the DG method, employs finite elements discretizations in space and time simultaneously with bilinear basis functions that are continuous in space and discontinuous in time. Based on using the Zienkiewicz-Zhu error estimate in space and the jumps (discontinuities) of displacements and velocities in the total energy norm as a local error estimate in time, an h -adaptive procedure which updates the spatial mesh and the time step size automatically so as to control the estimated errors within specified tolerances is proposed. Numerical examples are presented, showing that the considered DG method is of second-order accuracy in space (in L 2 ) and third-order accuracy in time, and the adaptive procedure is capable of updating the spatial meshes and the time steps when necessary, making the solutions reliable and the computation efficient.

STRUCTURAL DYNAMIC ANALYSIS BY A TIME‐DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD

This paper studies a time-discontinuous Galerkin finite element method for structural dynamic problems, by which both displacements and velocities are approximated as piecewise linear functions in the time domain and may be discontinuous at the discrete time levels. A new iterative solution algorithm which involves only one factorization for each fixed time step size and a few iterations at each step is presented for solving the resulted system of coupled equations. By using the jumps of the displacements and the velocities in the total energy norm as error indicators, an adaptive time-stepping procedure for selecting the proper time step size is described. Numerical examples including both single-DOF and multi-DOF problems are used to illustrate the performance of these algorithms. Comparisons with the exact results and/or the results by the Newmark integration scheme are given. It is shown that the time-discontinuous Galerkin finite element method discussed in this study possesses good accuracy (third order) and stability properties, its numerical implementation is not difficult, and the higher computational cost needed in each time step is compensated by use of a larger time step size.

Adaptive finite element procedures for linear and non‐linear dynamics

This paper discusses implementation and adaptivity of the Discontinuous Galerkin (DG) finite element method as applied to linear and non-linear structural dynamic problems. By the DG method, both displacements and velocities are approximated as piecewise bilinear functions in space and time and may be discontinuous at the discrete time levels. Both implicit and explicit iterative algorithms for solving the resulted system of coupled equations are derived. They are third-order accurate and, while the implicit procedure is unconditionally stable, the explicit one is conditionally stable. An h-adaptive procedure based on the Zienkiewicz–Zhu error estimate using the SPR technique is applied. Numerical examples are presented to show the suitability of the DG method for both linear and non-linear structural dynamic analysis. Copyright

Wave Propagation Related to High-Speed Train - a Scaled Boundary FE-approach for Unbounded Domains

Abstract Analysis of wave propagation in solid materials under moving loads is a topic of great interest in railway engineering. The objective of the present paper is three-dimensional modelling of high-speed train related ground vibrations; in particular the question of how to account for the unbounded media is addressed. For efficient and accurate modelling of railway structural components taking the unbounded media into account, a hybrid method based on the combination of conventional finite element method and scaled boundary finite element method is established. In the paper, element matrices and solution procedures for the Scaled Boundary Finite Element Method (SBFEM) are derived. A non-linear finite element iteration scheme using Lagrange multipliers and coupling between the unbounded domain and the finite element domain are also discussed. Two numerical examples including one example demonstrating the dynamical response of a railroad section are presented to demonstrate the performance of the proposed method.

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.

A posteriori error estimate by element patch post-processing, adaptive analysis in energy and L2 norms

Abstract This paper first presents a post-processing technique for obtaining a posteriori error estimators both in the energy norm and in the L 2 norm. For each element, an element patch , which represents the union of the considered element and its neighbours, is introduced. The post-processing for determining more accurate solutions is made by fitting a higher order polynomial expansion to the finite element solutions at superconvergent points in the patch by the least squares method. The element error estimate norms are calculated directly from the improved solutions. Another topic is the h -version adaptive finite element analysis for 2D linear elastic problems by coupling the error estimators with a mesh generator. T3 and T6 elements with the energy norm and a T6 element with the L 2 norm are used. Two examples, including a model for which exact solutions are available and a gravity dam under water pressure are presented. Numerical results show that the element patch post-processing provides asymptotically exact error estimates and the adaptive procedure produces finite element solutions with specified accuracy efficiently and economically.

Error estimation and adaptive procedures based on superconvergent patch recovery (SPR) techniques

SummaryThe paper discusses error estimation and adaptive finite element procedures for elasto-static and dynamic problems based on superconvergent patch recovery (SPR) techniques. The SPR is a postprocessing procedure to obtain improved finite element solutions by the least squares fitting of superconvergent stresses at certain sampling points in local patches. An enhancement of the original SPR by accounting for the equilibirum equations and boundary conditions is proposed. This enhancement improves the quality of postprocessed solutions considerably and thus provides an even more effective error estimate. The patch configuration of SPR can be either the union of elements surrounding a vertex node, thenode patch, or, the union of elements surrounding an element, theelement patch. It is shown that these two choices give normally comparable quality of postprocessed solutions. The paper is also concerned with the application of SPR techniques to a wide range of problems. The plate bending problem posted in mixed form where force and displacement variables are simultaneously used as unknowns is considered. For eigenvalue problems, a procedure of improving eigenpairs and error estimation of the eigenfrequency is presented. A postprocessed type of error estimate and an adaptive procedure for the semidiscrete finite element method are discussed. It is shown that the procedure is able to update the spatial mesh and the time step size so that both spatial and time discretization errors are controlled within specified tolerances. A discontinuous Galerkin method for solving structural dynamics is also presented.

Adaptive h-version eigenfrequency analysis

In dynamics it is important to compute natural frequencies and modes with a demanded accuracy. This paper presents an adaptive h-version finite element scheme to control the discretization error in free vibration analysis. Crucial parts of an adaptive analysis are the error estimator and the error indicator of the discretization error. The error indicator is based on the energy norm of the error. The estimate that constitutes both the error estimator and the error indicator are founded on post-processed eigenmodes in the absence of exact eigenmodes. The post-processing technique used is a mix of local and global updating methods. The local updating is based on the superconvergent patch recovery technique for displacements (SPRD). This approach provides reliable results for a sufficiently fine finite element mesh due to its strong dependence on the original finite element solution. In to overcome the necessity for meshes fine enough, a preconditioned conjugate gradient method is applied for the global updating using p+1 finite element formulation. The preconditioning matrix is implemented as a diagonal of the sum of stiffness and mass matrices in order to increase computational efficiency and the rate of convergence of the iterative procedure. Rapid convergence is enhanced by choosing the initial trial eigenmodes as the SPRD improved finite element eigenmodes. Finally we improve the global updated solution by applying the SPRD technique one more time. Numerical examples show the nice properties of the final local and global updated solution as a basis for an error estimator and an error indicator in an adaptive process.

Improved eigenfrequencies and eigenmodes in free vibration analysis

Abstract This paper presents an application of local and global updating methods to improve natural frequencies and mode shapes of the finite element solution in free vibration analysis. The local updating is based upon superconvergent patch recovery technique for displacements (SPRD). This approach provides reliable results for a sufficiently fine finite element mesh for the reason that it strongly depends on the original finite element solution. In order to overcome aforementioned shortcoming a preconditioned conjugate gradient method is applied for the global updating using p +1 finite element formulation. To increase the rate of convergence of the iterative procedure, a diagonal of the sum of stiffness and mass matrices is implemented as a preconditioning matrix in the general eigenvalue problem and initial trial eigenvectors are obtained by the SPRD technique. Numerical examples show for the eigenfrequencies an improved accuracy and an improved convergence rate of the error of the improved eigenfrequencies compared to the FE solution.