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.     

A Class of Mathematical Programs with Equilibrium Constraints: A Smooth Algorithm and Applications to Contact Problems

We discuss a special mathematical programming problem with equilibrium constraints (MPEC), that arises in material and shape optimization problems involving the contact of a rod or a plate with a rigid obstacle. This MPEC can be reduced to a nonlinear programming problem with independent variables and some dependent variables implicity defined by the solution of a mixed linear complementarity problem (MLCP). A projected-gradient algorithm including a complementarity method is proposed to solve this optimization problem. Several numerical examples are reported to illustrate the efficiency of this methodology in practice.     

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.