New analytical benchmarks for topology optimization and their implications

Sokó? and Rozvany (Struct Multidisc Optim doi:10.1007/s00158-012-0786-4, 2012) have published a very remarkable paper showing benchmarks solutions for the so named “two load problems”. Although the Writer strongly believes that the Authors have reached the very best solutions, he thinks that some details of the paper are worth of a few comments and questions. As a matter of fact, the Writer is personally concerned as his own benchmarks for this problem—from his PhD. dissertation of 1994—have been at last improved by those of the Authors.     

Local stability of trusses in the context of topology optimization Part I: Exact modelling

The paper considers the problem of optimal truss topology design with respect to stress and local stability (i.e. buckling) constraints. In a context of topology optimization, the exact. management of buckling constraints is highly complex: member forces must satisfy functions which discontinuously depend on the design variables.New terminologies and an exact problem formulation are provided. It turns out that the classical constraints (equilibrium, stress) together with topological local buckling constraints do not necessarily guarantee the existence of a solution structure. We discuss a simple but typical example demonstrating this effect inherently contained in the problem. It is proved that the inclusion of slenderness constraints guarantees a solution. These additional constraints are motivated by practice and preserve the topology nature of the problem. Finally, an alternative formulation is developed serving as a basis for computational approaches. The numerical treatment is the topic of Part II.     

Exact least-volume trusses for two symmetric point loads and unequal permissible stresses in tension and compression

In this Authors’ Reply previously presented solutions are extended to unequal permissible stresses in tension and compression, comments are made on the lack of adjoint displacement fields in so-called O-regions (memberless regions), and some computational details are explained.     

Exact analytical solutions for some popular benchmark problems in topology optimization II: three-sided polygonal supports

In an earlier paper (Rozvany, Struct Optim 15:42–48, 1998), the second author summarized known analytical solutions for some popular benchmark problems in topology optimization. In this, and in some subsequent papers, further exact optimal topologies are derived for least-weight, stress-controlled trusses, with load and support conditions that are frequently used in benchmark examples for numerical methods in topology optimization.     

A new solution for topology optimization problems with multiple loads:The guide-weight method

The guide-weight method is introduced to solve two kinds of topology optimization problems with multiple loads in this paper.The guide-weight method and its Lagrange multipliers' solution methods are presented first,and the Lagrange multipliers' soution method of problems with multiple constraints is improved by the dual method.Then the iterative formulas of the guide-weight method for topology optimization problems of minimum compliance and minimum weight are derived and coresponding numerical examples are...     

Exact analytical solutions for non-selfadjoint variable-topology shape optimization problems: perforated cantilever plates in plane stress subject to displacement constraints. Part I

Lurie (1994, 1995a, b) proved recently that variabletopology shape optimization of perforated plates in flexure for non-selfadjoint problems leads to rank-2 microstructures which are in general nonorthogonal. An extension of the same optimal microstructures to perforated plates in plane stress will be presented in Part II of this study. Using the above microstructure, the optimal solution is derived in this part for cantilever plates in plane stress, which are subject to two displacement constraints. For low volume fractions the above solutions are shown to converge to the known truss solutions of Birkeret al. (1994). The problem of homogenizing the stiffness of nonorthogonal rank-2 microstructures is also discussed.     

A review of particle swarm optimization. Part I: background and development

Particle Swarm Optimization (PSO), in its present form, has been in existence for roughly a decade, with formative research in related domains (such as social modelling, computer graphics, simulation and animation of natural swarms or flocks) for some years before that; a relatively short time compared with some of the other natural computing paradigms such as artificial neural networks and evolutionary computation. However, in that short period, PSO has gained widespread appeal amongst researchers and has been shown to offer good performance in a variety of application domains, with potential for hybridisation and specialisation, and demonstration of some interesting emergent behaviour. This paper aims to offer a compendious and timely review of the field and the challenges and opportunities offered by this welcome addition to the optimization toolbox. Part I discusses the location of PSO within the broader domain of natural computing, considers the development of the algorithm, and refinements introduced to prevent swarm stagnation and tackle dynamic environments. Part II considers current research in hybridisation, combinatorial problems, multicriteria and constrained optimization, and a range of indicative application areas.     

The Genesis distributed-memory benchmarks. Part 1: Methodology and general relativity benchmark with results for the SUPRENUM computer

This is the first of a series of papers on the Genesis distributed-memory benchmarks, which were developed under the European ESPRIT research program. The benchmarks provide a standard reference Fortran77 uniprocessor version, a distributed memory. MIMD version, and in some cases a Fortran90 version suitable for SIMD computers. The problems selected all have a scientific origin (mostly from physics or theoretical chemistry), and range from synthetic code fragments designed to measure the basic hardware properties of the computer (especially communication and synchronisation overheads), through commonly used library subroutines, to full application codes. This first paper defines the methodology to be used to analyse the benchmark results, and gives an example of a fully analysed application benchmark from General Relativity (GR1). First, suitable absolute performance metrics are carefully defined, then the performance analysis treats the execution time and absolute performance as functions of at least two variables, namely the problem size and the number of proecssors. The theoretical predictions are compared with, or fitted to, the measured results, and then used to predict (with due caution) how the performance might scale for larger problems and more processors than were actually available during the benchmarking. Benchmark measurements are given primarily for the German SUPRENUM computer, but also for the IBM 3083J, Convex C210 and a Parsys Supernode with 32 T800-20 transputers.     

Move limits definition in structural optimization with sequential linear programming. Part I: Optimization algorithm

A variety of numerical methods have been proposed in literature in purpose to deal with the complexity and non-linearity of structural optimization problems. In practical design, sequential linear programming (SLP) is very popular because of its inherent simplicity and because linear solvers (e.g. Simplex) are easily available. However, SLP performance is sensitive to the definition of proper move limits for the design variables which task itself often involves considerable heuristics. This research presents a new SLP algorithm (LESLP--linearization error sequential linear programming) that implements an advanced technique for defining the move limits. The LESLP algorithm is formulated so to overcome the traditional limitations of the SLP method. The new algorithm is successfully tested in weight minimization problems of truss structures with up to hundreds of design variables and thousands of constraints: sizing and configuration problems are considered. Optimization problems of non-truss structures are also presented.The key-ideas of LESLP and the discussion on numerical efficiency of the new algorithm are presented in a two-part paper. The first part concerns the basics of the LESLP formulation and provides potential users with a guide to programming LESLP on computers. In a companion paper, the numerical efficiency, advantages and drawbacks of LESLP are discussed and compared to those of other SLP algorithms recently published or implemented in commercial software packages.     

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.     

Modeling Multibody Systems with Uncertainties. Part I: Theoretical and Computational Aspects

This study explores the use of generalized polynomial chaos theory for modeling complex nonlinear multibody dynamic systems in the presence of parametric and external uncertainty. The polynomial chaos framework has been chosen because it offers an efficient computational approach for the large, nonlinear multibody models of engineering systems of interest, where the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction). The proposed methodology allows the quantification of uncertainty distributions in both time and frequency domains, and enables the simulations of multibody systems to produce results with “error bars”. The first part of this study presents the theoretical and computational aspects of the polynomial chaos methodology. Both unconstrained and constrained formulations of multibody dynamics are considered. Direct stochastic collocation is proposed as less expensive alternative to the traditional Galerkin approach. It is established that stochastic collocation is equivalent to a stochastic response surface approach. We show that multi-dimensional basis functions are constructed as tensor products of one-dimensional basis functions and discuss the treatment of polynomial and trigonometric nonlinearities. Parametric uncertainties are modeled by finite-support probability densities. Stochastic forcings are discretized using truncated Karhunen-Loeve expansions. The companion paper “Modeling Multibody Dynamic Systems With Uncertainties. Part II: Numerical Applications” illustrates the use of the proposed methodology on a selected set of test problems. The overall conclusion is that despite its limitations, polynomial chaos is a powerful approach for the simulation of multibody systems with uncertainties.     

New behavioral‐level simulation technique for RF/microwave applications. Part I: Basic concepts

The quadrature modeling technique is nowadays widely used for the nonlinear simulation of RF/microwave communication circuits and systems at the behavioral (system) level. It allows one to simulate the circuit/system performance under real‐world conditions and signals (using several thousand sample frequencies) and to predict such parameters as adjacent channel power ratio, spectral regrowth, and error vector magnitude in a computationally efficient way. But it is a narrowband technique and, consequently, cannot predict harmonics of the carrier frequency and even‐order nonlinear products, to account for the circuit/system frequency response and the bias decoupling network effect. Here, we propose a new behavioral‐level simulation technique (instantaneous quadrature technique) that overcomes these drawbacks, and demonstrate its validity through measurements and harmonic balance simulation. The transformation of envelope transfer characteristics into instantaneous ones is also discussed in detail. © 2000 John Wiley & Sons, Inc. Int J RF and Microwave CAE 10: 221–237, 2000.     

Data mining methods for knowledge discovery in multi-objective optimization: Part B - New developments and applications

The first part of this paper served as a comprehensive survey of data mining methods that have been used to extract knowledge from solutions generated during multi-objective optimization. The current paper addresses three major shortcomings of existing methods, namely, lack of interactiveness in the objective space, inability to handle discrete variables and inability to generate explicit knowledge. Four data mining methods are developed that can discover knowledge in the decision space and visualize it in the objective space. These methods are (i) sequential pattern mining, (ii) clustering-based classification trees, (iii) hybrid learning, and (iv) flexible pattern mining. Each method uses a unique learning strategy to generate explicit knowledge in the form of patterns, decision rules and unsupervised rules. The methods are also capable of taking the decision maker’s preferences into account to generate knowledge unique to preferred regions of the objective space. Three realistic production systems involving different types of discrete variables are chosen as application studies. A multi-objective optimization problem is formulated for each system and solved using NSGA-II to generate the optimization datasets. Next, all four methods are applied to each dataset. In each application, the methods discover similar knowledge for specified regions of the objective space. Overall, the unsupervised rules generated by flexible pattern mining are found to be the most consistent, whereas the supervised rules from classification trees are the most sensitive to user-preferences.     

Halogenated ligands and their interactions with amino acids: implications for structure-activity and structure-toxicity relationships

The properties of chemicals are rooted in their molecular structure. It follows that structural analysis of specific interactions between ligands and biomolecules at the molecular level is invaluable for defining structure-activity relationships (SARs) and structure-toxicity relationships (STRs). This study has elucidated the structural and molecular basis of interactions of biomolecules with alkyl and aryl halides that are extensively used as components in many commercial pesticides, disinfectants, and drugs. We analyzed the protein structures deposited in Protein Data Bank (PDB) for structural information associated with interactions between halogenated ligands and proteins. This analysis revealed distinct patterns with respect to the nature and structural characteristics of halogen interactions with specific types of atoms and groups in proteins. Fluorine had the highest propensity of interactions for glycine, while chlorine for leucine, bromine for arginine, and iodine for lysine. Chlorine, bromine and iodine had the lowest propensity of interactions for cysteine, while fluorine had a lowest propensity for proline. These trends for highest propensity shifted towards the hydrophobic residues for all the halogens when only interactions with the side chain were considered. Halogens had equal propensities of interaction for the halogen bonding partners (nitrogen and oxygen atoms), albeit with different geometries. The optimal angle for interactions with halogens was approximately 120 degrees for oxygen atoms, and approximately 96 degrees for nitrogen atoms. The distance distributions of halogens with various amino acids were mostly bimodal, and the angle distributions were unimodal. Insights gained from this study have implications for the rational design of safer drugs and commercially important chemicals.     

Linear stability analysis in fluid-structure interaction with transpiration. Part I: Formulation and mathematical analysis

The aim of this work is to provide a new Linearization Principle approach particularly suited for problems in fluid-structure stability. The complexity here, and the main difference with respect to the classical approach, comes from the fact that the full non-linear fluid equations are written in a moving (i.e. time dependent) domain. The underlying idea of our approach uses transpiration techniques [J. Fluid Mech. 4 (1958) 383; G. Mortchéléwicz, Application of linearized Euler equations to flutter, in: 85th AGARD SMP Meeting, Aalborg, Denmark, 1997; P. Raj, B. Harris, Using surface transpiration with an Euler method for cost-effective aerodynamic analysis, in: AIAA 24th Applied Aerodynamics Conference, number 93-3506, Monterey, Canada, 1993; AIAA 27(6) (1989) 777], with the formalization and linearization recently developed in [Rév. Européenne Élém. Finis, 9(6-7) (2000) 681, A. Dervieux (Ed.), Fluid-Structure Interaction, Kogan Page Science, London, 2003 (Chapter 3)]. This allows us to obtain a new grid independent coupled spectral problem involving the linearized Navier-Stokes equations and those of a reduced linear structure. The coupling is realized through specific transpiration conditions acting on a fixed interface, while keeping a fixed fluid domain. We provide a rigorous mathematical treatment of this eigenproblem. We prove that the corresponding eigenmodes, characterizing the free evolution of the system, can be obtained from the characteristic values of a compact operator acting on a Hilbert space. Moreover, we localize the eigenfrequencies of the system in a parabolic region of the complex plan centered along the positive real axis.