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.     

Topology optimization of cellular materials with periodic microstructure under stress constraints

Material design is a critical development area for industries dealing with lightweight construction. Trying to respond to these industrial needs topology optimization has been extended from structural optimization to the design of material microstructures to improve overall structural performance. Traditional formulations based on compliance and volume control result in stiffness-oriented optimal designs. However, strength-oriented designs are crucial in engineering practice. Topology optimization with stress control has been applied mainly to (macro) structures, but here it is applied to material microstructure design. Here, in the context of density-based topology optimization, well-established techniques and analyses are used to address known difficulties of stress control in optimization problems. A convergence analysis is performed and a density filtering technique is used to minimize the risk of results inaccuracy due to coarser finite element meshes associated with highly non-linear stress behavior. A stress-constraint relaxation technique (qp-approach) is applied to overcome the singularity phenomenon. Parallel computing is used to minimize the impact of the local nature of the stress constraints and the finite difference design sensitivities on the overall computational cost of the problem. Finally, several examples test the developed model showing its inherent difficulties.

     

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.     

Optimal topology design for stress-isolation of soft hyperelastic composite structures under imposed boundary displacements

Soft hyperelastic composite structures that integrate soft hyperelastic material and linear elastic hard material can undergo large deformations while isolating high strain in specified locations to avoid failure. This paper presents an effective topology optimization-based methodology for seeking the optimal united layout of hyperelastic composite structures with prescribed boundary displacements and stress constraints. The optimization problem is modeled based on the power-law interpolation scheme for two candidate materials (one is soft hyperelastic material and the other is linear elastic material). The ?-relaxation technique and the enhanced aggregation method are employed to avoid stress singularity and improve the computational efficiency. Then, the topology optimization problem can be readily solved by a gradient-based mathematical programming algorithm using the adjoint variable sensitivity information. Numerical examples are given to show the importance of considering prescribed boundary displacements in the design of hyperelastic composite structures. Moreover, numerical solutions demonstrate the validity of the present model for the optimal topology design with a stress-isolated region.     

On body shapes providing maximum depth of penetration

The problem of maximization of the depth of penetration of rigid impactor into semi-infinite solid media (concrete shield) is investigated analytically and numerically using two-stage model and experimental data of Forrestal and Tzou (Int J Solids Struct 34(31–32):4127–4146, 1997). The shape of the axisymmetric rigid impactor has been taken as an unknown design variable. To solve the formulated optimization problem for nonadditive functional, we expressed the depth of penetration (DOP) under some isoperimetric constraints. We apply approaches based on analytical and qualitative variational methods and numerical optimization algorithm of global search. Basic attention for considered optimization problem was given to constraints on the mass of penetrated bodies, expressed by the volume in the case of penetrated solid body and by the surface area in the case of penetrated thin-walled rigid shell. As a result of performed investigation, based on two-term and three-term two stage models proposed by Forrestal et al. (Int J Impact Eng 15(4):396–405, 1994), Forrestal and Tzou (Int J Solids Struct 34(31–32):4127–4146, 1997) and effectively developed by Ben-Dor et al. (Comp Struct 56:243–248, 2002, Comput Struct 81(1):9–14, 2003a, Int J Solids Struct 40(17):4487–4500, 2003b, Mech Des Struct Mach 34(2): 139–156, 2006), we found analytical and numerical solutions and analyzed singularities of optimal forms.     

Solving steel mill slab design problems

The steel mill slab design problem from the CSPLIB is a combinatorial optimization problem motivated by an application of the steel industry. It has been widely studied in the constraint programming community. Several methods were proposed to solve this problem. A steel mill slab library was created which contains 380 instances. A closely related binpacking problem called the multiple knapsack problem with color constraints, originated from the same industrial problem, was discussed in the integer programming community. In particular, a simple integer program for this problem has been given by Forrest et al. (INFORMS J Comput 18:129–134, 2006). The aim of this paper is to bring these different studies together. Moreover, we adapt the model of Forrest et al. (INFORMS J Comput 18:129–134, 2006) for the steel mill slab design problem. Using this model and a state-of-the-art integer program solver all instances of the steel mill slab library can be solved efficiently to optimality. We improved, thereby, the solution values of 76 instances compared to previous results (Schaus et al., Constraints 16:125–147, 2010). Finally, we consider a recently introduced variant of the steel mill slab design problem, where within all solutions which minimize the leftover one is interested in a solution which requires a minimum number of slabs. For that variant we introduce two approaches and solve all instances of the steel mill slab library with this slightly changed objective function to optimality.     

On globally stable singular truss topologies

We consider truss topology optimization problems including a global stability constraint, which guarantees a sufficient elastic stability of the optimal structures. The resulting problem is a nonconvex semi-definite program, for which nonconvex interior point methods are known to show the best performance. We demonstrate that in the framework of topology optimization, the global stability constraint may behave similarly to stress constraints, that is, that some globally optimal solutions are singular and cannot be approximated from the interior of the design domain. This behaviour, which may be called a global stability singularity phenomenon, prevents convergence of interior point methods towards globally optimal solutions. We propose a simple perturbation strategy, which restores the regularity of the design domain. Further, to each perturbed problem interior point methods can be applied.     

Note on singular optima in laminate design problems

This paper studies the design of laminates subject to restrictions on the ply strength. The minimum weight design is considered. It is shown that this formulation includes singular optima, which are similar to the ones observed in topology optimization including local stress constraints. In laminate design, these singular optima are linked to the removal of ‘zero thickness’ plies from the stacking sequence. It is shown how the fiber orientation variables can circumvent the singularity by relaxing the strength constraints related to such vanishing plies. This demonstrates the key role of fiber orientations in the optimization of laminates and the need for their efficient treatment as design variables.     

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 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.     

Solving stress constrained problems in topology and material optimization

This article is a continuation of the paper Ko?vara and Stingl (Struct Multidisc Optim 33(4?C5):323?C335, 2007). The aim is to describe numerical techniques for the solution of topology and material optimization problems with local stress constraints. In particular, we consider the topology optimization (variable thickness sheet or ??free sizing??) and the free material optimization problems. We will present an efficient algorithm for solving large scale instances of these problems. Examples will demonstrate the efficiency of the algorithm and the importance of the local stress constraints. In particular, we will argue that in certain topology optimization problems, the addition of stress constraints must necessarily lead not only to the change of optimal topology but also optimal geometry. Contrary to that, in material optimization problems the stress singularity is treated by the change in the optimal material properties.     

Layout design of reinforced concrete structures using two-material topology optimization with Drucker–Prager yield constraints

This paper aims to develop a method that can automatically generate the optimal layout of reinforced concrete structures by incorporating concrete strength constraints into the two-material topology optimization formulation. The Drucker–Prager yield criterion is applied to predict the failure behavior of concrete. By using the power-law interpolation, the proposed optimization model is stated as a minimum compliance problem under the yield stress constraints on concrete elements and the material volume constraint of steel. The ε-relaxation technique is employed to prevent the stress singularity. A hybrid constraint-reduction strategy, in conjunction with the adjoint-variable sensitivity information, is integrated into a gradient-based optimization algorithm to overcome the numerical difficulties that arise from large-scale constraints. It can be concluded from numerical investigations that the proposed model is suitable for obtaining a reasonable layout which makes the best uses of the compressive strength of concrete and the tensile strength of steel. Numerical results also reveal that the hybrid constraint-reduction strategy is effective in solving the topology optimization problems involving a large number of constraints.     

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.     

A new approach for the solution of singular optima in truss topology optimization with stress and local buckling constraints

The present paper investigates problems of truss topology optimization under local buckling constraints. A new approach for the solution of singular problems caused by stress and local buckling constraints is proposed. At first, a second order smooth-extended technique is used to make the disjoint feasible domains connect, then the so-called ε-relaxed method is applied to eliminate the singular optima from problem formulation. By means of this approach, the singular optimum of the original problem caused by stress and local buckling constraints can be searched approximately by employing the algorithms developed for sizing optimization problems with high accuracy. Therefore, the numerical problem resulting from stress and local buckling constraints can be solved in an elegant way. The applications of the proposed approach and its effectiveness are illustrated with several numerical examples. Received May 2, 2000     

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.     

Petri net-based context modeling for context-aware systems

The context-aware services require to efficiently perceive not only the user requirements but also the context of the environment to provide customized services to the user. To efficiently develop the context-aware applications a systematic methodology correctly specifying the relation among dynamically changing contexts is essential. Here the context model simplifying the manipulation of complex contexts is a key accessor for the specification and analysis of the service. Among various modeling approaches such as timed automata (Tang and You in Intell Automat Soft Comput 16(4):605–619, 2010), workflow (Rosemann et al. in Understanding context-awareness in business process design, 2010), Petri net (PN) (J?rgensen et al. in Innovat Syst Softw Eng 5(1):13–25, 2009), etc. developed for context-aware system, the PN-based approach has been recognized as one of the most effective one. In this paper we identify the issues of how the contexts are modeled and what kinds of the requirements needs to be considered in the context processing. We then discuss various Petri net (PN)-based modeling methodologies concerning the five important features for context processing: relationships and dependencies, time constraint, resource constraint, usability of modeling formalisms, and context identification. The study reveals that the approach effectively allowing both the time and resource constraints in the model while supporting other important properties needs to be developed further for accurately assess the context-aware systems. Also, the expandability and scalability issue need to be investigated.     

Topology optimization for microstructural design under stress constraints

This work aims at introducing stress responses within a topology optimization framework applied to the design of periodic microstructures. The emergence of novel additive manufacturing techniques fosters research towards new approaches to tailor materials properties. This paper derives a formulation to prevent the occurrence of high stress concentrations, often present in optimized microstructures. Applying macroscopic test strain fields to the material, microstructural layouts, reducing the stress level while exhibiting the best overall stiffness properties, are sought for. Equivalent stiffness properties of the designed material are predicted by numerical homogenization and considering a metallic base material for the microstructure, it is assumed that the classical Von Mises stress criterion remains valid to predict the material elastic allowable stress at the microscale. Stress constraints with arbitrary bounds are considered, assuming that a sizing optimization step could be applied to match the actual stress limits under realistic service loads. Density–based topology optimization, relying on the SIMP model, is used and the qp–approach is exploited to overcome the singularity phenomenon arising from the introduction of stress constraints with vanishing material. Optimization problems are solved using mathematical programming schemes, in particular MMA, so that a sensitivity analysis of stress responses at the microstructural level is required and performed considering the adjoint approach. Finally, the developed method is first validated with classical academic benchmarks and then illustrated with an original application: tailoring metamaterials for a museum anti–seismic stand.     

A randomized algorithm for the on-line weighted bipartite matching problem

We study the on-line minimum weighted bipartite matching problem in arbitrary metric spaces. Here, n not necessary disjoint points of a metric space M are given, and are to be matched on-line with n points of M revealed one by one. The cost of a matching is the sum of the distances of the matched points, and the goal is to find or approximate its minimum. The competitive ratio of the deterministic problem is known to be Θ(n), see (Kalyanasundaram, B., Pruhs, K. in J. Algorithms 14(3):478–488, 1993) and (Khuller, S., et al. in Theor. Comput. Sci. 127(2):255–267, 1994). It was conjectured in (Kalyanasundaram, B., Pruhs, K. in Lecture Notes in Computer Science, vol. 1442, pp. 268–280, 1998) that a randomized algorithm may perform better against an oblivious adversary, namely with an expected competitive ratio Θ(log n). We prove a slightly weaker result by showing a o(log ³ n) upper bound on the expected competitive ratio. As an application the same upper bound holds for the notoriously hard fire station problem, where M is the real line, see (Fuchs, B., et al. in Electonic Notes in Discrete Mathematics, vol. 13, 2003) and (Koutsoupias, E., Nanavati, A. in Lecture Notes in Computer Science, vol. 2909, pp. 179–191, 2004). The authors were partially supported by OTKA grants T034475 and T049398.     

Approximation Algorithms for Soft-Capacitated Facility Location in Capacitated Network Design

Answering an open question published in Operations Research (54, 73–91, 2006) in the area of network design and logistic optimization, we present the first constant-factor approximation algorithms for the problem combining facility location and cable installation in which capacity constraints are imposed on both facilities and cables. We study the problem of designing a minimum cost network to serve client demands by opening facilities for service provision and installing cables for service shipment. Both facilities and cables have capacity constraints and incur buy-at-bulk costs. This Max SNP-hard problem arises in diverse applications and is shown in this paper to admit a combinatorial 19.84-approximation algorithm of cubic running time. This is achieved by an integration of primal-dual schema, Lagrangian relaxation, demand clustering and bi-factor approximation. Our techniques extend to several variants of this problem, which include those with unsplitable demands or requiring network connectivity, and provide constant-factor approximate algorithms in strongly polynomial time. X. Chen is Visiting Fellow, University of Warwick.     

Reliability-based topology optimization of continuum structures subject to local stress constraints

Topology optimization of continuum structures is a challenging problem to solve, when stress constraints are considered for every finite element in the mesh. Difficulties are compounding in the reliability-based formulation, since a probabilistic problem needs to be solved for each stress constraint. This paper proposes a methodology to solve reliability-based topology optimization problems of continuum domains with stress constraints and uncertainties in magnitude of applied loads considering the whole set of local stress constrains, without using aggregation techniques. Probabilistic constraints are handled via a first-order approach, where the principle of superposition is used to alleviate the computational burden associated with inner optimization problems. Augmented Lagrangian method is used to solve the outer problem, where all stress constraints are included in the augmented Lagrangian function; hence sensitivity analysis may be performed only for the augmented Lagrangian function, instead of for each stress constraint. Two example problems are addressed, for which crisp black and white topologies are obtained. The proposed methodology is shown to be accurate by checking reliability indices of final topologies with Monte Carlo Simulation.