AN OVERVIEW OF RECENT ADVANCES AND EVALUATION OF THE VIRTUAL-PULSE (VIP) TIME-INTEGRAL METHODOLOGY FOR GENERAL STRUCTURAL DYNAMICS PROBLEMS: COMPUTATIONAL ISSUES AND IMPLEMENTATION ASPECTS

An overview of new and recent advances towards a VIrtual-Pulse (VIP) time-integral methodology for general linear/non-linear dynamic systems is presented. Attention is focused on providing a brief overview and an indepth evaluation of the developments, methodology, computational issues and implementation aspects for practical problems. Different from the way we have been looking at the developments encompassing existing direct time-integration type methods and mode superposition techniques, the proposed methodology capitalizes on the computational attributes of both and thereby offers new perspectives and several attractive favourable features in terms of stability and accuracy, storage, and computational costs for a wide variety of inertial dynamic problems. Recently, the authors have shown the theoretical developments via the VIP methodology for transient structural problems and for transient thermal problems. The purpose of the present paper is to summarize the theoretical developments, improve upon the computational/implementation aspects for general linear/non-linear dynamic structural problems, and demonstrate the pros and cons via numerous illustrative test cases. The theoretical analysis and results of several test cases show that the VIP methodology has improved accuracy/stability characteristics and computational advantages in comparison to the commonly advocated explicit and implicit methods such as the Newmark family. Overall, an analysis of the theoretical developments, algorithmic study, and the implementation and evaluation of the formulations strongly suggest the proposition that the VIP methodology is a viable alternative for general structural dynamic applications encountered in practical engineering problems.     

Error-controlled adaptive mixed finite element methods for second-order elliptic equations

In this contribution, we deal with a posteriori error estimates and adaptivity for mixed finite element discretizations of second-order elliptic equations, which are applied to the Poisson equation. The method proposed is an extension to the one recently introduced in [10] to the case of inhomogeneous Dirichlet and Neumann boundary conditions. The residual-type a posteriori error estimator presented in this paper relies on a postprocessed and therefore improved solution for the displacement field which can be computed locally, i.e. on the element level. Furthermore, it is shown that this discontinuous postprocessed solution can be further improved by an averaging technique. With these improved solutions at hand, both upper and lower bounds on the finite element discretization error can be obtained. Emphasis is placed in this paper on the numerical examples that illustrate our theoretical results.     

UNCONDITIONALLY STABLE HIGHER-ORDER ACCURATE HERMITIAN TIME FINITE ELEMENTS

In this paper, single step time finite elements using the cubic Hermitian shape functions to interpolate the solution over a time interval are considered. The second-order differential equations are manipulated directly. Both the effects of modal damping and external excitation are considered. The accuracy of the solutions at the end of the time interval and the interpolated solutions within the time interval is investigated. The weighted residual approach is adopted to derive the time-integration algorithms. Instead of specifying the weighting functions, the weighting parameters are used to control the characteristics of the time finite elements. The weighting parameters are chosen to eliminate the higher-order truncation error terms or to enforce the asymptotic annihilation condition. A one-parameter family of third-order accurate asymptotically annihilating algorithms and another one-parameter family of fourth-order accurate non-dissipative algorithms are presented. The ranges of the weighting parameters for unconditionally stable algorithms are given. It is found that one of the members in each family corresponds to the Padé approximants of the exponential function in solving the first-order differential equations. Some of the existing unconditionally stable higher-order accurate algorithms are re-derived by the present unified approach.     

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.     

<Emphasis Type="Italic">p</Emphasis>-Version error estimation for linear elasticity

An error estimator, formulated earlier for h-adaptive strategies, is extended for use in the p-version finite element analysis. The estimation of error is based on solving a series of local problems, based on patches consisting of elements surrounding each node, with prescribed homogeneous essential boundary conditions. Unlike the original approach in which a patch was constructed based on one element, each patch in the present scheme is automatically formed based on a number of elements surrounding a corresponding node. The present scheme, based on enhancing the degree of interpolation, provides a better estimate than the original h-scheme while still preserving the original lower bound property. The capability of the new scheme is investigated in some numerical examples in terms of its global and local performance.     

An effective data parallel self-starting explicit methodology for computational structural dynamics on the connection machine CM-5

This paper discusses the implementation aspects and our experiences towards a data parallel explicit self-starting finite element transient methodology with emphasis on the Connection Machine (CM-5) for linear and non-linear computational structural dynamic applications involving structured and unstructured grids. The parallel implementation criteria that influence the efficiency of an algorithm include the amount of communication, communication routing, and load balancing. To provide simplicity, high level of accuracy, and to retain the generality of the finite element implementation for both linear and non-linear transient explicit problems on a data parallel computer which permit optimum amount of communications, we implemented the present self-starting dynamic formulations (in comparison to the traditional approaches) based on nodal displacements, nodal velocities, and elemental stresses on the CM-5. Data parallel language CMFortran is employed with virtual processor constructs and with:SERIAL and:PARALLEL layout directives for arrays. The communications via the present approach involve only one gather operation (extraction of element nodal displacements or velocities from global displacement vector) and one scatter operation (dispersion of element forces onto global force vector) for each time step. These gather and scatter operations are implemented using the Connection Machine Scientific Software Library communication primitives for both structured and unstructured finite element meshes. The implementation aspects of the present self-starting formulations for linear and elastoplastic applications on serial and data parallel machines are discussed. Numerical test models for linear and non-linear one-dimensional applications and a two-dimensional unstructured finite element mesh are then illustrated and their performance studies are discussed.