Substructuring Methods for Computing the Nullspace of Equilibrium Matrices

Abstract

Equations of equilibrium arise in numerous areas of engineering. Applications to electrical networks, structures, and fluid flow are elegantly described in Introduction to Applied Mathematics, Wellesley Cambridge Press, Wellesley, MA, 1986 by Strang. The context in which equilibrium equations arise may be stated in two forms: Constrained Minimization Form: $\min(x^T Ax - 2x^T r)\,{\text{subject to}}\,Ex = s$, Lagrange Multiplier Form: $EA^{ - 1} E^T \lambda = s - EA^{ - 1} r\,{\text{and}}\,Ax = r - E^T \lambda $. The Lagrange multiplier form given above results from block Gaussian elimination on the $2 \times 2$ block matrix system for the constrained minimization form. Here A is generally some symmetric positive-definite matrix associated with the minimization problem. For example, A can be the element flexibility matrix in the structures application. An important approach (called the force method in structural optimization) to the solution to such problems involves dimension reduction nullspace schemes based upon computation of a basis for the nullspace for E. In our approach to solving such problems we emphasize the parallel computation of a basis for the nullspace of E and examine the applications to structural optimization and fluid flow. Several new block decomposition and node ordering schemes are suggested and reanalyses computations are investigated. Comparisons of these schemes are made with those of Storaasli et al. for structures and Hall et al. for fluids.