George I. N. Rozvany

The COC algorithm, Part II: Topological, geometrical and generalized shape optimization

Abstract After outlining analytical methods for layout optimization and illustrating them with examples, the COC algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members. The numerical results obtained are shown to be in close agreement (up to twelve significant digits) with analytical results. Finally, the problem of generalized shape optimization (finding the best boundary topology and shape) is discussed.

Generalized shape optimization without homogenization

Two types of solutions may be considered in generalized shape optimization. Absolute minimum weight solutions, which are rather unpractical, consist of solid, empty and porous regions. In more practical solutions of shape optimization, porous regions are suppressed and only solid and empty regions remain. This note discusses this second class of problems and shows that a solid, isotropic microstructure with an adjustable penalty for intermediate densities is efficient in generating optimal topologies.

Optimal Layout of Grillages

ABSTRACT The paper deals with the minimum-weight design of elastic grillages in which the absolute value of the axial stresses nowhere exceeds a prescribed value. After discussing optimality conditions for this problem, a geometrical method for obtaining the optimal beam directions is presented. Considering quadrilateral grillages, the effect of translation and rotation of sides on the optimal layout is investigated. Finally, the concept of “beam weaves” is introduced and the optimal layout of beams near a free edge is discussed.

Extended exact solutions for least-weight truss layouts—Part I: Cantilever with a horizontal axis of symmetry

Abstract The paper deals with new applications of Michells theory for determining the layout of least-weight trusses under a single load condition and the same permissible stress for all bars. Considering cantilever trusses with a horizontal axis of symmetry, the optimal layout was derived in a closed analytical form by A. S. L. Chan and H. S. Y. Chan for a limited length: height ratio. Solutions beyond the above ratio are investigated in this paper. The results will be extended to non-symmetric cantilever trusses in Part II of this study. In both parts, a novel method is used for deriving adjoint displacements. The results are also confirmed by comparisons with discretized solutions for trusses and perforated plates.

Optimization of large structural systems

Principal lectures: optimality criteria and topology optimization decomposition methods and approximation concepts sensitivity analysis mathematical programming and global optima composite, anisotropic and non-linear materials neural networks, parallel processing, multicriteria and control problems miscellaneous topics abstracts of contributed papers.

On the Solid Plate Paradox in Structural Optimization

ABSTRACT This paper discusses the minimum weight design of solid plastic plates subject to constraints on the highest and lowest allowable values of the plate thickness. It is shown that the maximum thickness constraint alone does not ensure a smooth global minimum weight solution because the least weight is furnished, in the limit, by a grillage-like continuum consisting of a dense system of ribs of infinitesimal spacing and uniform depth. The optimal layout of such continua has already been determined for most loading and boundary conditions by the first author and Prager and is found to be the same for (a) plastic limit design, (b) elastic stress design, (c) design for given compliance, and (d) design for given fundamental frequency. Two refinements of the above layout theory are also considered in the current paper. One formulation takes into consideration the weight savings due to rib intersections in high density grillages. The other development deals with minimum as well as maximum thickness constr...

On singular topologies in exact layout optimization

The causes of singular structural topologies, which prevent most iterative computational algorithms from reaching the global optimal solution, are explained in the light of the theory of exact optimal layouts. This theory is also used for deriving eight fundamental characteristics of singular topologies. The above findings are illustrated with case studies of exact optimal layouts for a single load and for two load conditions with stress constraints.

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

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.

Michell layouts for various combinations of line supports—I

Abstract The optimal layout of least-weight grillages or beam systems of given depth has been determined in a closed analytical form for almost all conceivable boundary and load conditions. In spite of its early introduction at the turn of the century, the theory of least-weight trusses or “Michell-frameworks” has yielded, on the other hand, only few solutions for relatively restricted supports and loadings. The aim of this paper is to explore Michell layouts, when supports are provided anywhere along given lines. It is shown that for this less restricted case the solutions correspond to fields of constant strain, which give layouts consisting of a finite number of straight members. More complex classes of solutions will be discussed in a separate paper.

Grillages of maximum strength and maximum stiffness

Abstract Assuming a preassigned beam depth, minimum weight solutions are derived for both perfectly plastic and elastic grillages of given strength as well as elastic grillages of given stiffness. The solutions presented also give a minimum reinforcement volume for perfectly plastic fibre-reinforced plates. Common kinematic optimality conditions are stated for the foregoing classes of problems and a method is outlined for finding the optimal solution for any clamped boundary. Morleys claim regarding the non-existence of certain kinematically admissible optimal solutions is shown to be erroneous. The proposed technique is illustrated with a number of examples.