Developments in structural analysis by direct energy minimization.

The equivalence of problems in structural analysis to certain variational statements has traditionally been used as a means of formulating field equations and natural boundary conditions. In recent years developments in computers and minimization algorithms have made it possible to determine the displacement state that satisfies the variational principle without direct recourse to the solution of a system of equations. In this paper an unconstrained minimization approach is applied to a problem in finite-element structural analysis and is shown numerically to be competitive with conventional methods. Adopting an energyminimization viewpoint, important storage advantages of the conjugate-gradient method are extended by eliminating the need for an assembled stiffness matrix. The convergence of conjugate direction methods in quadratic problems is shown to be greatly influenced by scale effects. This problem is studied using the eigenvalues of the stiffness matrix and a practical solution is proposed which by a special scaling transformation can improve the ratio of the maximum to the minimum eigenvalue by several orders of magnitude. This transformation is applied with considerable success to a plate-bending problem.