Topology optimization of 2D continua for minimum compliance using parallel computing

Topology optimization is often used in the conceptual design stage as a preprocessing tool to obtain overall material distribution in the solution domain. The resulting topology is then used as an initial guess for shape optimization. It is always desirable to use fine computational grids to obtain high-resolution layouts that minimize the need for shape optimization and postprocessing (Bendsoe and Sigmund, Topology optimization theory, methods and applications. Springer, Berlin Heidelberg New York 2003), but this approach results in high computation cost and is prohibitive for large structures. In the present work, parallel computing in combination with domain decomposition is proposed to reduce the computation time of such problems. The power law approach is used as the material distribution method, and an optimality criteria-based optimizer is used for locating the optimum solution [Sigmund (2001)21:120–127; Rozvany and Olhoff, Topology optimization of structures and composites continua. Kluwer, Norwell 2000]. The equilibrium equations are solved using a preconditioned conjugate gradient algorithm. These calculations have been done using a master–slave programming paradigm on a coarse-grain, multiple instruction multiple data, shared-memory architecture. In this study, by avoiding the assembly of the global stiffness matrix, the memory requirement and computation time has been reduced. The results of the current study show that the parallel computing technique is a valuable tool for solving computationally intensive topology optimization problems.     

Constrained mesh optimization on boundary

This paper presents a new mesh optimization approach aiming to improve the mesh quality on the boundary. The existing mesh untangling and smoothing algorithms (Vachal et al. in J Comput Phys 196: 627–644, 2004; Knupp in J Numer Methods Eng 48: 1165–1185, 2002), which have been proved to work well to interior mesh optimization, are enhanced by adding constrains of surface and curve shape functions that approximate the boundary geometry from the finite element mesh. The enhanced constrained optimization guarantees that the boundary nodes to be optimized always move on the approximated boundary. A dual-grid hexahedral meshing method is used to generate sample meshes for testing the proposed mesh optimization approach. As complementary treatments to the mesh optimization, appropriate mesh topology modifications, including buffering element insertion and local mesh refinement, are performed in order to eliminate concave and distorted elements on the boundary. Finally, the optimization results of some examples are given to demonstrate the effectivity of the proposed approach.     

On an alternative approach to stress constraints relaxation in topology optimization

The paper deals with the imposition of local stress constraints in topology optimization. The aim of the work is to analyze the performances of an alternative methodology to the ε-relaxation introduced in Cheng and Guo (Struct Optim 13:258–266, 1997), which handles the well-known stress singularity problem. The proposed methodology consists in introducing, in the SIMP law used to apply stress constraints, suitable penalty exponents that are different from those that interpolate stiffness parameters. The approach is similar to the classical one because its main effect is to produce a relaxation of the stress constraints, but it is different in terms of convergence features. The technique is compared with the classical one in the context of stress-constrained minimum-weight topology optimization. Firstly, the problem is studied in a modified truss design framework, where the arising of the singularity phenomenon can be easily shown analytically. Afterwards, the analysis is extended to its natural context of topology bidimensional problems.     

Comment on “Topology optimization of energy harvesting devices using piezoelectric materials”

In a recent paper, Zheng et al. (Struct Multidisc Optim 38:17–23, 2009) presented a topology optimization formulation for the design of energy harvesting devices considering the maximization of a predefined energy conversion factor. Their motivation was based on the observed trend of using piezoelectric effects to build energy harvesting devices or to harvest electrical energy from ambient vibrations. This discussion addresses some unclear points in the mentioned work.     

Efficient topology optimization in MATLAB using 88 lines of code

The paper presents an efficient 88 line MATLAB code for topology optimization. It has been developed using the 99 line code presented by Sigmund (Struct Multidisc Optim 21(2):120–127, 2001) as a starting point. The original code has been extended by a density filter, and a considerable improvement in efficiency has been achieved, mainly by preallocating arrays and vectorizing loops. A speed improvement with a factor of 100 is obtained for a benchmark example with 7,500 elements. Moreover, the length of the code has been reduced to a mere 88 lines. These improvements have been accomplished without sacrificing the readability of the code. The 88 line code can therefore be considered as a valuable successor to the 99 line code, providing a practical instrument that may help to ease the learning curve for those entering the field of topology optimization. The paper also discusses simple extensions of the basic code to include recent PDE-based and black-and-white projection filtering methods. The complete 88 line code is included as an appendix and can be downloaded from the web site .     

A discrete level-set topology optimization code written in Matlab

This paper presents a compact Matlab implementation of the level-set method for topology optimization. The code can be used to minimize the compliance of a statically loaded structure. Simple code modifications to extend the code for different and multiple load cases are given. The code is inspired by a Matlab implementation of the solid isotropic material with penalization (SIMP) method for topology optimization (Sigmund, Struct Multidiscipl Optim 21:120–127, 2001). Including the finite element solver and comments, the code is 129 lines long. The code is intended for educational purposes, and in particular it could be used alongside the Matlab implementation of the SIMP method for topology optimization to demonstrate the similarities and differences between the two approaches.     

A 199-line Matlab code for Pareto-optimal tracing in topology optimization

The paper ‘A 99-line topology optimization code written in Matlab’ by Sigmund (Struct Multidisc Optim 21(2):120–127, 2001) demonstrated that SIMP-based topology optimization can be easily implemented in less than hundred lines of Matlab code. The published method and code has been used even since by numerous researchers to advance the field of topology optimization. Inspired by the above paper, we demonstrate here that, by exploiting the notion of topological-sensitivity (an alternate to SIMP), one can generate Pareto-optimal topologies in about twice the number of lines of Matlab code. In other words, optimal topologies for various volume fractions can be generated in a highly efficient manner, by directly tracing the Pareto-optimal curve.     

Some Properties of Digital Covering Spaces

In this paper, we study digital versions of some properties of covering spaces from algebraic topology. Among our results are some that correct or improve upon the presentation of assertions in earlier papers (Boxer and Karaca in J. Math. Imaging Vis. 32:23–29, 2008; Han in Inf. Sci. 177:3731–3748, 2007; Han in Inf. Sci. 178:550–561, 2008).     

On some applications of ant colony optimization metaheuristic to plane truss optimization

Ant colony optimization metaheuristic (ACO) represents a new class of algorithms particularly suited to solve real-world combinatorial optimization problems. ACO algorithms, published for the first time in 1991 by M. Dorigo [Optimization, learning and natural algorithms (in Italian). Ph.D. Thesis, Dipartimento di Elettronica, Politecnico di Milano, Milan, 1992] and his coworkers, have been applied, particularly starting from 1999 (Bonabeau et al., Swarm intelligence: from natural to artificial systems, Oxford University Press, New York, 1999; Dorigo et al., Artificial life 5(2):137–172, 1999; Dorigo and Di Caro, Ant colony optimization: a new metaheuristic, IEEE Press, Piscataway, NJ, 1999; Dorigo et al., Ant colony optimization and swarm intelligence, Springer, Berlin Heidelberg New York, 2004; Dorigo and Stutzle, Ant colony optimization, MIT Press, Cambridge, MA, 2004), to several kinds of optimization problems such as the traveling salesman problem, quadratic assignment problem, vehicle routing, sequential ordering, scheduling, graph coloring, management of communications networks, and so on. The ant colony optimization metaheuristic takes inspiration from the studies of real ant colonies’ foraging behavior. The main characteristic of such colonies is that individuals have no global knowledge of problem solving but communicate indirectly among themselves, depositing on the ground a chemical substance called pheromone, which influences probabilistically the choice of subsequent ants, which tend to follow paths where the pheromone concentration is higher. Such behavior, called stigmergy, is the basic mechanism that controls ant activity and permits them to take the shortest path connecting their nest to a food source. In this paper, it is shown how to convert natural ant behavior to algorithms able to escape from local minima and find global minimum solutions to constrained combinatorial problems. Some examples on plane trusses are also presented.     

Exact analytical solutions for some popular benchmark problems in topology optimization III: <Emphasis Type="Italic">L</Emphasis>-shaped domains

In this paper exact, analytical solutions are derived for another highly popular benchmark problem, namely, L-shaped domains having a horizontal line support and one or several point loads. The optimal topologies are obtained in the context of Michell structures, i.e., least-weight, stress, or compliance-controlled trusses with a single load condition.     

An Automatic Correction of Ma’s Thinning Algorithm Based on <Emphasis Type="Italic">P</Emphasis>-simple Points

The notion of P-simple points was introduced by Bertrand to conceive parallel thinning algorithms. In ‘A 3D fully parallel thinning algorithm for generating medial faces’ (Pattern Recogn. Lett. 16:83–87, 1995), Ma proposed an algorithm for which there are objects whose topology is not preserved. In this paper, we propose a new application of P-simple points: to automatically correct Ma’s algorithm.     

Explicit level set and density methods for topology optimization with equivalent minimum length scale constraints

The goal of this paper is to introduce local length scale control in an explicit level set method for topology optimization. The level set function is parametrized explicitly by filtering a set of nodal optimization variables. The extended finite element method (XFEM) is used to represent the non-conforming material interface on a fixed mesh of the design domain. In this framework, a minimum length scale is imposed by adopting geometric constraints that have been recently proposed for density-based topology optimization with projections filters. Besides providing local length scale control, the advantages of the modified constraints are twofold. First, the constraints provide a computationally inexpensive solution for the instabilities which often appear in level set XFEM topology optimization. Second, utilizing the same geometric constraints in both the density-based topology optimization and the level set optimization enables to perform a more unbiased comparison between both methods. These different features are illustrated in a number of well-known benchmark problems for topology optimization.

     

EM323: a line search based algorithm for solving high-dimensional continuous non-linear optimization problems

This paper presents a performance study of a one-dimensional search algorithm for solving general high-dimensional optimization problems. The proposed approach is a hybrid between a line search algorithm of Glover (The 3-2-3, stratified split and nested interval line search algorithms. Research report, OptTek Systems, Boulder, CO, 2010) and an improved variant of a global method of Gardeux et al. (Unidimensional search for solving continuous high-dimensional optimization problems. In: ISDA ’09: Proceedings of the 2009 ninth international conference on intelligent systems design and applications, IEEE Computer Society, Washington, DC, USA, pp 1096–1101, 2009) that uses line search algorithms as subroutines. The resulting algorithm, called EM323, was tested on 19 scalable benchmark functions, with a view to observing how optimization techniques for continuous optimization problems respond with increasing dimension. To this end, we report the algorithm’s performance on the 50, 100, 200, 500 and 1,000-dimension versions of each function. Computational results are given comparing our method with three leading evolutionary algorithms. Statistical analysis discloses that our method outperforms the other methods by a significant margin.     

Risk-neutral evaluation of information security investment on data centers

Based on given data center network topology and risk-neutral management, this work proposes a simple but efficient probability-based model to calculate the probability of insecurity of each protected resource and the optimal investment on each security protection device when a data center is under security breach. We present two algorithms that calculate the probability of threat and the optimal investment for data center security respectively. Based on the insecurity flow model (Moskowitz and Kang 1997) of analyzing security violations, we first model data center topology using two basic components, namely resources and filters, where resources represent the protected resources and filters represent the security protection devices. Four basic patterns are then identified as the building blocks for the first algorithm, called Accumulative Probability of Insecurity, to calculate the accumulative probability of realized threat (insecurity) on each resource. To calculate the optimal security investment, a risk-neutral based algorithm, called Optimal Security Investment, which maximizes the total expected net benefit is then proposed. Numerical simulations show that the proposed approach coincides with Gordon’s (Gordon and Loeb, ACM Transactions on Information and Systems Security 5(4):438–457, 2002) single-system analytical model. In addition, numerical results on two common data center topologies are analyzed and compared to demonstrate the effectiveness of the proposed approach. The technique proposed here can be used to facilitate the analysis and design of more secured data centers.     

A note on the alternating group network

The class of alternating group networks was introduced in the late 1990’s as an alternative to the alternating group graphs as interconnection networks. Recently, additional properties for the alternating group networks have been published. In particular, Zhou et al., J. Supercomput (2009), doi:, was published very recently in this journal. We show that this so-called new interconnection topology is in fact isomorphic to the (n,n−2)-star, a member of the well-known (n,k)-stars, 1≤k≤n−1, a class of popular networks proposed earlier for which a large amount of work have already been done. Specifically, the problem in Zhou et al., J. Supercomput (2009), doi:, was addressed in Lin and Duh, Inf. Sci. 178(3), 788–801, 2008, when k = n−2.     

Analytical derivatives technology for structural shape design

The ability to perform and evaluate the effect of shape changes on the stress and modal responses of components is an important ingredient in the “design” of aircraft engine components. The classical design of experiments (DOE)-based approach that is motivated from statistics (for physical experiments) is one of the possible approaches for the evaluation of the component response with respect to design parameters [Myers, Montgomery. Response surface methodology, process and product optimization using design of experiments. John Wiley and Sons, NY (1995)]. As the underlying physical model used for the component response is deterministic and understood through a computer simulation model, one needs to re-think the use of the classical DOE techniques for this class of problems. In this paper, we explore an alternate sensitivity-analysis-based technique where a deterministic parametric response is constructed using exact derivatives of the complex finite-element (FE)-based computer models to design parameters. The method is based on a discrete sensitivity analysis formulation using semi-automatic differentiation (Griewank, SIAM (2000), ADIFOR, Automatic Differentiation of FORTRAN codes ) to compute the Taylor series or its Pade equivalent for finite-element-based responses. Shape design or optimization in the context of finite element modeling is challenging because the evaluation of the response for different shape requires the need for a meshing consistent with the new geometry. This paper examines the differences in the nature and performance (accuracy and efficiency) of the analytical derivatives approach against other existing approaches with validation on several benchmark structural applications. The use of analytical derivatives for parametric analysis is demonstrated to have accuracy benefits on certain classes of shape applications.     

A fuzzy particle swarm optimization algorithm for computer communication network topology design

Particle swarm optimization (PSO) is a powerful optimization technique that has been applied to solve a number of complex optimization problems. One such optimization problem is topology design of distributed local area networks (DLANs). The problem is defined as a multi-objective optimization problem requiring simultaneous optimization of monetary cost, average network delay, hop count between communicating nodes, and reliability under a set of constraints. This paper presents a multi-objective particle swarm optimization algorithm to efficiently solve the DLAN topology design problem. Fuzzy logic is incorporated in the PSO algorithm to handle the multi-objective nature of the problem. Specifically, a recently proposed fuzzy aggregation operator, namely the unified And-Or operator (Khan and Engelbrecht in Inf. Sci. 177: 2692–2711, 2007), is used to aggregate the objectives. The proposed fuzzy PSO (FPSO) algorithm is empirically evaluated through a preliminary sensitivity analysis of the PSO parameters. FPSO is also compared with fuzzy simulated annealing and fuzzy ant colony optimization algorithms. Results suggest that the fuzzy PSO is a suitable algorithm for solving the DLAN topology design problem.     

Exact methods for the resource levelling problem

We present a project scheduling problem subject to general temporal constraints where the utilization of a set of renewable resources has to be smoothed over time. For solving the NP-hard optimization problem, we point out some important structural properties and introduce a new enumeration scheme. Combining this enumeration scheme with some branch-and-bound techniques, we propose an appropriate solution procedure for the project scheduling problem at hand. To outline the practical importance of resource levelling, we sketch exemplary the optimization of mid-term planning schedules with regard to the resource requirements of IT projects. Finally, we present results from a comprehensive computational study on problem instances of the well-known rlp_j10 and rlp_j20 test sets devised by Kolisch et al. (Benchmark Instances for Project Scheduling Problems, Kluwer, Boston, 1999).     

Michell cantilevers constructed within trapezoidal domains—Part I: geometry of Hencky nets

The present paper is the first part of the four-part work on Michell cantilevers transmitting a given point load to a given segment of a straight-line support, the feasible domain being a part of the half-plane contained between two fixed half-lines. The axial stress σ in the optimal cantilevers is assumed to be bounded by −σ _C ≤σ≤σ _T , where σ _C and σ _T represent the allowable compressive and tensile stresses, respectively. The work provides generalization of the results of the article of Lewiński et al. (Int J Mech Sci 36:375–398, 1994a) to the case of σ _T ≠σ _C . The present, first part of the work concerns the analytical formation of the Hencky nets or the lines of fibres filling up the interior of the optimal cantilevers corresponding to an arbitrary position of the point of application of the given concentrated force.     

Some notes on maximal arc intersection of spherical polygons: its \mathcal{NP} -hardness and approximation algorithms

Finding a sequence of workpiece orientations such that the number of setups is minimized is an important optimization problem in manufacturing industry. In this paper we present some interesting notes on this optimal workpiece setup problem. These notes show that (1) The greedy algorithm proposed in Comput. Aided Des. 35 (2003), pp. 1269–1285 for the optimal workpiece setup problem has the performance ratio bounded by O(ln n−ln ln n+0.78), where n is the number of spherical polygons in the ground set; (2) In addition to greedy heuristic, linear programming can also be used as a near-optimal approximation solution; (3) The performance ratio by linear programming is shown to be tighter than that of greedy heuristic in some cases.