Anna Feriani

Iterative system solvers for the frequency analysis of linear mechanical systems

Abstract The paper deals with the numerical treatment of the direct frequency domain (DFD) analysis of linear mechanical systems. Attention is mainly focused on the solution of the complex system of equations needed for each analyzed frequency. Strategies for transforming the system into the “shifted” form ( T −σ I ) z = d are proposed and discussed, where σ is related to the frequency. For two formulations of the shifted system the performance of some Krylov sub-space iterative solvers is tested and compared to that of a multi-frontal direct method. Advantage is taken of the shifted form in solving simultaneously a large number of systems resulting from different values of the shift (frequency). Numerical experiments on some prototype structural dynamics problems are reported; the results shown demonstrate how the devised strategies for the iterative solution can outperform, in many cases, the direct solver.

The formation of viscous damping matrices for the dynamic analysis of MDOF systems

The paper deals with the representation of dissipative effects by means of equivalent viscous forces. A brief review of the classical analytical treatment of the subject is first presented devoting particular attention to the topics of hysteretic and modal damping. The problem of forming viscous matrices in the case of systems which are non-homogeneous from the point of view of dissipation is then addressed; soil-structure systems are first considered and some accepted techniques for forming the structure contribution to the viscous matrix are reviewed. A different technique is then proposed which avoids some of the drawbacks of the previously quoted methods. In the final section of the paper it is shown how this technique is easily applicable also in the case of systems having internal concentrated dampers of viscous type, this being a situation which is difficult to tackle with usual criteria.

An incremental elastic-plastic Finite Element solver in a workstation cluster environment Part I. Formulations and parallel processing☆

Abstract We discuss solution schemes for the incremental elastic-plastic structural problem, discretized by means of the Finite Element method. Attention is focused on their formulation and implementation in a parallel computing environment defined by a cluster of workstations connected by means of a network. The availability of parallel computers allows one to consider possible formulations and solution strategies so far not considered competitive with the classical Newton-like schemes implying the definition of an elastic-plastic tangent stiffness matrix. The solution strategies herein considered are based on the explicit integration of the actual elastic-plastic rate problem. This, in turn, is phrased in terms of two different formulations, whose relative advantages—particularly with respect to their integration in parallel—are discussed. A u − gl (displacemen plastic multiplier) formulation of the structural rate theory of plasticity [1], integrated by means of an explicit, element-by-element scheme, seems to be the most promising one.

An incremental elastic-plastic Finite Element solver in a workstation cluster environment. Part II. Performance of a first implementation

Abstract This work illustrates results obtained by implementing in a parallel computer environment the gl rate formulation of the theory of plasticity and its integration scheme as illustrated in the preceding Part I. The Preconditioned Conjugate Gradient is used as a tool for repeatedly solving linear systems of equations. Although the investigation about the performance of the workstation cluster as a parallel virtual computer is still far from being completed, it is already possible to conclude that, in such an environment, several techniques proposed for reducing the computer time required by the iterative solver are not applicable. One purpose of the work is therefore to give guidelines in terms of expected performances of the Conjugate Gradient method when applied to stiff problems, in which the condition number may shoot up to the billions. Even if the implemented computer code is not based on the most convenient rate formulation in terms of parallelizability, as shown in Part I of this work, the obtained results indicate anyway that, for some categories of structural problems, the development of a parallel, element-by-element computer code is a promising line of work.