On the (non-)optimality of Michell structures

Optimal analytical Michell frame structures have been extensively used as benchmark examples in topology optimization, including truss, frame, homogenization, density and level-set based approaches. However, as we will point out, partly the interpretation of Michell’s structural continua as discrete frame structures is not accurate and partly, it turns out that limiting structural topology to frame-like structures is a rather severe design restriction and results in structures that are quite far from being stiffness optimal. The paper discusses the interpretation of Michell’s theory in the context of numerical topology optimization and compares various topology optimization results obtained with the frame restriction to cases with no design restrictions. For all examples considered, the true stiffness optimal structures are composed of sheets (2D) or closed-walled shell structures (3D) with variable thickness. For optimization problems with one load case, numerical results in two and three dimensions indicate that stiffness can be increased by up to 80 % when dropping the frame restriction. For simple loading situations, studies based on optimal microstructures reveal theoretical gains of +200 %. It is also demonstrated how too coarse design discretizations in 3D can result in unintended restrictions on the design freedom and achievable compliance.     

Michell structures formed on surfaces of revolution

The paper formulates the Michell-like problems for surface gridworks. Particular attention is devoted to the problem of designing the lightest fully stressed gridworks formed on surfaces of revolution. In the examples considered, the gridworks are subjected to torsion. Proof is given that the circular meridian is a minimizer of the weight (or volume) functional of a shell subjected to torsion, thus justifying the original Michell conjecture according to which just the spherical twisting shell is the lightest. The proof is based on the methods of the classical variational calculus and thus can be viewed as elementary. This result is confirmed by a direct comparison of the exact formulae for the weight of a spherical Michell shell with the exact formulae for the weights of optimal conical and cylindrical shells with the same fixed boundaries.     

On nonunique solutions in topology optimization

This paper deals with the problem of non-unique solutions in topology optimization. Depending on the optimization path, the solutions, in other words the topologies of a structure, are different. The nonuniqueness problem in topology optimization is presented in connection with the testing of different lower material mass value bounding functions and the use of different material properties updating functions and different threshold functions. The structure strain energy minimum criterion is applied to find the optimum topology. A comparison of the topologies obtained from the energy criterion point of view is made.     

A strain tensor method for three-dimensional Michell structures

In the design of discrete structures such as trusses and frames, important quantitative goals such as minimal weight or minimal compliance often dominate. Many numerical techniques exist to address these needs. However, an analytical approach exists to meet similar goals, which was initiated by Michell (1904) and has been mostly used for two-dimensional structures so far. This paper develops a method to extend the existing mainly two-dimensional approach to apply to three-dimensional structures. It will be referred as the Michell strain tensor method (MSTM). First, the proof that MSTM is consistent with the existing theory in two dimensions is provided. Second, two- and three-dimensional known solutions will be replicated based on MSTM. Finally, MSTM will be used to solve new three-dimensional cases.     

A general boundary approach to the construction of Michell truss structures

The goal of the present study is to provide a building block approach which will enable the synthesis of new Michell truss structure solutions. Curved support boundaries for Michell truss structures are categorized into four types. Each type is graphically illustrated as a simple example structure. A general matrix operator method is developed to solve the layout of each type. Numerical solutions that use the matrix method are compared with analytical solutions in a case study that comprises complimentary logarithmic spirals on a circular arc boundary. To illustrate the applications of this study, numerical layout solutions on a circular support boundary are explored that produce a family of globally-optimal Michell cantilever solutions for the support of a distributed load along a straight cantilever flange.     

Variational proof of optimality criteria for Michell structures with pre-existing members

This note puts forward a direct variational derivation of the optimality criteria of a class of Michell-like structures in which some members are a priori given; an alternative derivation was published in Rozvany et al. (Struct Multidisc Optim 31:373–377, 2006).     

Partial relaxation of the orthogonality requirement for classical Michell trusses

It is often stated, even in standard references, that in classical Michell trusses (i.e. least-weight trusses for one load condition with a stress or compliance constraint) a pair of intersecting compression and tensile bars must always be orthogonal. The aim of this brief note is to show that there are important exceptions to this rule and that the modification of this restriction enables us to obtain new classes of solutions.     

Analytic solutions of elastically supported Michell trusses

Structural and Multidisciplinary Optimization - In this short note, analytic solutions of elastically supported Michell truss with varying support stiffness and support positions are presented....     

On the convergence of validity interval analysis

Validity interval analysis (VIA) is a generic tool for analyzing the input-output behavior of feedforward neural networks. VIA is a rule extraction technique that relies on a rule refinement algorithm. The rules are of the form R(i)-->R(0) i.e. "if the input of the neural network is in the region R(i), then its output is in the region R(0)," where regions are axis parallel hypercubes. VIA conjectures, then refines and checks rules for inconsistency. This process can be computationally expensive, and the rule refinement phase becomes critical. Hence, the importance of knowing the complexity of these rule refinement algorithms. In this paper, we show that the rule refinement part of VIA always converges in one run for single-weight-layer networks, and has an exponential average rate of convergence for multilayer networks. We also discuss some variations of the standard VIA formulae.     

Stochastic approximation algorithms for the local optimization of functions with nonunique stationary points

The aim of this paper is the provision of a framework for a practical stochastic unconstrained optimization theory. The results are based on certain concepts of stochastic approximation, although not restricted to those procedures, and aim at incorporating the great flexibility of currently available deterministic optimization ideas into the stochastic problem, whenever optimization must be done by Monte Carlo or sampling methods. Hills with nonunique stationary points are treated. A framework has been provided, with which convergence of stochastic versions of conjugate gradient, partan, etc., can be discussed and proved.     

On the correlation between size and metric validity

Empirical validation of code metrics has a long history of success. Many metrics have been shown to be good predictors of external features, such as correlation to bugs. Our study provides an alternative explanation to such validation, attributing it to the confounding effect of size. In contradiction to received wisdom, we argue that the validity of a metric can be explained by its correlation to the size of the code artifact. In fact, this work came about in view of our failure in the quest of finding a metric that is both valid and free of this confounding effect. Our main discovery is that, with the appropriate (non-parametric) transformations, the validity of a metric can be accurately (with R-squared values being at times as high as 0.97) predicted from its correlation with size. The reported results are with respect to a suite of 26 metrics, that includes the famous Chidamber and Kemerer metrics. Concretely, it is shown that the more a metric is correlated with size, the more able it is to predict external features values, and vice-versa. We consider two methods for controlling for size, by linear transformations. As it turns out, metrics controlled for size, tend to eliminate their predictive capabilities. We also show that the famous Chidamber and Kemerer metrics are no better than other metrics in our suite. Overall, our results suggest code size is the only “unique” valid metric.     

On cluster validity for the fuzzy c-means model

Many functionals have been proposed for validation of partitions of object data produced by the fuzzy c-means (FCM) clustering algorithm. We examine the role a subtle but important parameter-the weighting exponent m of the FCM model-plays in determining the validity of FCM partitions. The functionals considered are the partition coefficient and entropy indexes of Bezdek, the Xie-Beni (1991), and extended Xie-Beni indexes, and the Fukuyama-Sugeno index (1989). Limit analysis indicates, and numerical experiments confirm, that the Fukuyama-Sugeno index is sensitive to both high and low values of m and may be unreliable because of this. Of the indexes tested, the Xie-Beni index provided the best response over a wide range of choices for the number of clusters, (2-10), and for m from 1.01-7. Finally, our calculations suggest that the best choice for m is probably in the interval [1.5, 2.5], whose mean and midpoint, m=2, have often been the preferred choice for many users of FCM     

On the safety of optimized structures

Confidence of designers in the theory of structural optimization is of fundamental importance for practical applications. The so-called erosion of optimal designs is discussed from this point of view, in the context of ring and plate examples for which the limit and post-limit behaviors are obtained either experimentally or by numerical simulations of experiments. Tentative conclusions are suggested.This paper was scheduled for presentation at the IUTAM Symposium in Melbourne, Feb. 1988, but the first author was unfortunately unable to attend     

On the dynamics of framed structures

The paper presents a comparison between the eigen-values of a structure obtained using the distributed mass-stiffness technique proposed by the author[l] and those obtained using a static stiffness matrix coupled with a mass matrix both of them based on the same shape function (consistent mass matrix). The paper also includes some suggestions for improving the results obtained using the latter method.     

Michell cantilevers constructed within a half strip. Tabulation of selected benchmark results

The paper delivers the benchmark results for the Michell cantilevers constructed within a half strip, for selected values of the σ _T /σ _C ratio, σ _T , σ _C being the admissible stresses in tension and compression, respectively.     

On the validity of ESO type methods in topology optimization

It is shown on a simple test example that ESO’s rejection criteria may result in a highly nonoptimal design. Reasons for this failure are also discussed. Received September 12, 2000     

On the dynamic response of lightweight structures

In this paper, the experimental determination of the dynamic response of lightweight structures is described. The experimental set-up, the required hardware system and some important points of the software system are described. Some results are displayed graphically and compared with computational results.     

Generalized Michell structures — exact least-weight truss layouts for combined stress and displacement constraints: Part I — General theory for plane trusses

In Part I of this study, earlier results are briefly reviewed and then general optimality criteria derived for exact leastweight plane truss layouts with combined stress and displacement constraints. Whilst these are necessary conditions for a local minimum with respect toany topology, Part II discusses analytical solutions within agiven two-bar topology for a vertical support and a point load. The latter results are used for verifying the general theory in Part I.     

On the aggregation of prioritized belief structures

The focus of this work is to provide a procedure for aggregating prioritized belief structures. Motivated by the ideas of nonmonotonic logics an alternative to the normalization step used in Dempster's rule when faced with conflicting belief structures is suggested. We show how this procedure allows us to make inferences in inheritance networks where the knowledge is in the form of a belief structure     

On Hoffman's celebrated cycling LP example

We answer two questions that naturally arise while dealing with Hoffman's celebrated 50-year-old linear program to be solved by the primal simplex method, where an angle θ$\theta$ and a scaling factor ω$\omega$ are adjustable parameters. In particular, we determine what conditions have to be imposed on ω$\omega$ for classical cycling to occur with θ=2π/5$\theta =2\pi /5$, and what on θ$\theta$ with ω=±tan(θ)$\omega =\pm \tan (\theta )$. The first answer reveals that the sufficient condition widely spread over the literature is false, so fixing it turns this example into a correct example of classical cycling. Some progress towards necessary and sufficient conditions for cycling to occur in this example is also reported.