Continuum-type optimality criteria methods for large finite element systems with a displacement constraint. Part I

Abstract

Continuum-type optimality criteria for the iterative optimization of “large” finite element systems (i.e. systems with over ten thousand finite elements) are discussed. By investigating optimization problems with up to one million elements and one million variables, it is shown that for a single displacement constraint the proposed method results in a rapid and almost uniform convergence, the rate of which, even for relatively ill-conditioned problems, does not depend significantly on the number of elements. Additional refinements, including upper and lower limits on the cross-sectional dimensions, segmentation, allowance for selfweight and the cost of supports, non-linear and non-separable objective functions, inclusion of shear deformations, built-up cross-sections as well as additional stress constraints and two-dimensional (plane stress) problems, will be considered in Part II of this contribution. The current development constitutes an extension and generalization of pioneering work by Berke, Khot, Venkayya and their associates, whose methods are also reviewed herein. In addition to an elementary truss example and a more advanced beam example, some simple layout optimization problems are considered in this Part. A special feature of the paper is that all numerical results presented are confirmed by closed form analytical solutions.