Substructure Algorithm Research Articles

To Overlap or Not to Overlap: A Note on a Domain Decomposition Method for Elliptic Problems

More than one hundred years ago, H.A. Schwarz introduced a domain decomposition method in which the original elliptic equation is solved on overlapping subregions, one after another, in an iterative process. A few years ago, Chan and Resasco, introduced a method that they classified as a domain decomposition method using nonoverlapping subdomains. In this note, it is shown that their method is an accelerated version of the classical method. It is also shown that the error propagation operator of the method can be expressed in terms of Schur complements of certain stiffness matrices and that techniques previously developed for the study of iterative substructuring algorithms can be used to derive estimates on the rate of convergence.

Multi-level substructuring in the elasto-plastic domain

The application of the multi-level substructuring technique to the analysis of elasto-plastic and local nonlinear problems is studied. After a review of principles and formulations a practical implementation and a typical flow chart of the procedure for the technique are presented. Two numerical examples are provided to demonstrate the efficiency that can be reached by using a multi-level substructuring algorithm.

On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations

Substructuring methods are in common use in structural mechanics problems where typically the associated linear systems of algebraic equations are positive definite. Here these methods are extended to problems which lead to nonpositive definite, nonsymmetric matrices. The extension is based on an algorithm which carries out the block Gauss elimination procedure without the need for interchanges even when a pivot matrix is singular. Examples are provided wherein the method is used in connection with finite element solutions of the stationary Stokes equations and the Helmholtz equation, and dual methods for second-order elliptic equations.

Concepts of a General Substructuring System for Structural Dynamics Analyses

Multilevel substructuring algorithms in structural dynamics are considered together with their software implementation. The paper emphasizes a consistent data structure for representing objects on hierarchical levels. The data structure permits recursive numerical procedures to be easily implemented. As an example, a recursive multi-subspace iteration algorithm is introduced. The paper also considers the definition and implementation of a problem specification language. The language is similar to and retains the expressive power of Lisp. Finally, a numerical example is given.

Substructuring techniques—status and projections

Status and recent developments of substructuring techniques and their application to structural analysis and design are summarized. Discussion focuses on a number of aspects including: multilevel substructuring algorithms; use of hypermatrix and other sparse matrix schemes, use of substructuring in automated design systems and application of substructuring in elasto-plastic problems. Numerical examples are presented to demoostrate the reduction in the number of arithmetic operations and disk storage requirements obtained by using multilevel substructuring techniques. Also discussed are the potential benefits of using such techniques with new computing hardware such as CDC STAR-100 and minicomputer systems.