Truss Topology Optimization Research Articles

Exploring new tensegrity structures via mixed integer programming

A tensegrity structure is a prestressed pin-jointed structure consisting of continuously connected tensile members (cables) and disjoint compressive members (struts). Many classical tensegrity structures are prestress stable, i.e., they are kinematically indeterminate but stabilized by introducing prestresses. This paper presents a procedure for generating various prestress stable tensegrity structures. This method is based on truss topology optimization and does not require connectivity relation of cables and struts of a tensegrity structure to be known in advance. Unlike the conventional form-finding methods, the locations of nodes are fixed throughout optimization. The optimization problem with the constraints expressing the definition of tensegrity structure, kinematical indeterminacy, and symmetry of configurations is formulated as a mixed integer linear programming (MILP) problem. Numerical experiments demonstrate that various tensegrity structures can be generated from one given initial structure by solving the presented MILP problems by using a few control parameters.

Topology optimization of trusses considering static and dynamic constraints using the CSS

Topology optimization of structures reveals outstanding advantages when compared to sectional optimization. Many unnecessary members and nodes may exist in a structure and a topology optimization provides an opportunity to remove them. This advantage will specially become apparent when comparatively large cost of the nodes is taken into account. Fundamental frequencies of a structure are important, easily obtained characteristics which allow the designer to keep out from the dangerous resonance phenomenon. When dynamic excitations are critical, these characteristics cannot be neglected. In this paper, topology optimization of truss structures is investigated considering stress, displacement, buckling and frequency constraints. To perform such an optimization is not simple because of large, highly nonlinear and non-convex search space. Here the newly developed charged system search algorithm is used to accomplish this optimization.

Multiobjective dynamic topology optimization of truss with interval parameters based on interval possibility degree

Static and dynamic multiobjective topology optimization of trusses with interval parameters is investigated. The uncertain parameters of the trusses are described by an interval model. The multiobjective topology optimization model of trusses with interval parameters is constructed. On the basis of Taylor expansion and natural interval extension, the stress and displacement response intervals under static loads and inherent natural frequency interval of truss are deduced. The non-deterministic optimization problem is transformed into a deterministic programming problem by minimizing maximum standards and the concept of interval possibility degree. The Pareto CMOPGA (genetic algorithm for constrained multiobjective optimization problem) embedding structural stability examination on the basis of ranking is adopted to solve the constrained multiobjective optimization problem. Two numerical examples show that the proposed method is effective and reasonable.

Topology optimization of trusses using bars exchange method

Abstract. The algorithm of optimization of trusses is presented in the paper, where for topology optimization the bars exchange method is used. In the first case, the problem aimed at cost minimization with a constraint set on global stiffness is formulated. In the second case, the problem of minimizing the cost function subjected to stress and cross-sectional area constraints is discussed and here the multiple-load case is taken into consideration. The conditions for introduction of topology modification and its acceptance are specified. The paper is illustrated with three examples.

Reinforcement layout design for concrete structures based on continuum damage and truss topology optimization

This article presents a new procedure for the layout design of reinforcement in concrete structures. Concrete is represented by a gradient-enhanced continuum damage model with strain-softening and reinforcement is modeled as elastic bars that are embedded into the concrete domain. Adjoint sensitivity analysis is derived in complete consistency with respect to path-dependency and the nonlocal model. Classical truss topology optimization based on the ground structure approach is applied to determine the optimal topology and cross-sections of the reinforcement bars. This approach facilitates a fully digital work flow that can be highly effective, especially for the design of complex structures. Several test cases involving two- and three-dimensional concrete structures illustrate the capabilities of the proposed procedure.

On the equivalence of minimum compliance and stress-constrained minimum weight design of trusses under multiple loading conditions

This article is devoted to topology optimization of trusses under multiple loading conditions. Compliance minimization with material volume constraint and stress-constrained minimum weight problem are considered. In the case of a single loading condition, it has been shown that the two problems have the same optimal topology. The possibility of extending this result for problems involving multiple loading conditions is examined in the present work. First, the compliance minimization problem is formulated as a multicriterion optimization problem, where the conflicting criteria are the compliances of the different loading conditions. Then, the optimal topologies of the stress-constrained minimum weight problem and the multicriterion compliance minimization problem for a simple test example are compared. The results verify that when multiple loading conditions are involved, the stress-constrained minimum weight topology cannot be obtained in general by solving the compliance minimization problem.

A Parellelzing Modified Particle Swarm Optimizer and its Application to Discrete Topological Optimization

recently, a modified Particle Swarm Optimizer (MLPSO) has been succeeded in solving truss topological optimization problems and competitive results are obtained. In order to reduce its execution time for solving large complex optimization problem, a parallel version for this optimizer (PMLPSO) is studied in this paper. This paper first gives an overview of PSO algorithm as well as the modified PSO, and then a design and an implementation of parallel PSO is proposed. Since most of structural problems involve discrete design variables, an effect strategy is involved in MLPSO in order to operate on discrete variables. The performance of the proposed algorithm is tested by two examples and promising speed-up rate is obtained. Final part is conclusion and outlook.

Exact truss topology optimization: allowance for support costs and different permissible stresses in tension and compression—extensions of a classical solution by Michell

The aim of this paper is twofold. It introduces a theory of Michell trusses taking support (reaction) costs into consideration, which is illustrated with a modified version of Michell's best known example. Then eight other variations on this example are presented, with (i) equal or unequal permissible stresses in tension and compression, (ii) the structural domain consisting of a half-plane or a full plane, and (iii) the supports being two pins, or a pin and a roller. Previously noted shortcomings of Michell's theory are highlighted and several fundamental properties of non-selfadjoint topology optimization problems discussed. The analytical solutions are verified to a high degree of accuracy by numerical results.

Truss topology optimization for mass and reliability considerations—co-evolutionary multiobjective formulations

The paper presents an approach for simultaneous optimization of structural mass and reliability in discrete truss structures. In addition to member sizing, the selection of an optimal topology from a pre-specified ground structure is a feature of the proposed methodology. To allow for a global search, optimization is performed using a multiobjective evolutionary algorithm. System reliability is based on a recently developed computational approach that is efficient and could be integrated within the framework of an evolutionary optimization process. The presence of multiple allowable topologies in the optimization process was handled through co-evolution in competing subpopulations. A unique feature of the algorithm is an automatic reunification of these populations using hypervolume measure-based indicator as reunification criterion to attain greater search efficiency. Numerical experiments demonstrate the computational advantages of the proposed method. These advantages become more pronounced for large-scale optimization problems, where the standard evolutionary approach fails to produce the desired results.

Modeling of User Design Preferences in Multiobjective Optimization of Roof Trusses

AbstractMany conceptual design programs determine optimality by relying on quantifiable design objectives such as minimizing weight and deflection. Qualitative design criteria related to architectural or constructability requirements, however, are not typically considered. This paper presents a preference-prediction model that evaluates user preference as an explicit design objective in the topology and geometry optimization of roof trusses. During optimization, each truss design is characterized by using a set of nine quantifiable features. A trained neural network inputs the characteristic truss feature values for each proposed design and outputs an integer value that indicates the user’s preference for that design. To reduce the number of user interactions required to collect the required neural network training data, a Kohonen self-organizing map clusters designs into groups on the basis of feature similarities. Clustering allows users to indicate preferences for groups of similar trusses instead of f...

Solution of a truss topology bilevel programming problem by means of an inexact restoration method

We formulate a truss topology optimization problem as a bilevel programming problem and solve it by means of a line search type inexact restoration algorithm. We discuss details of the implementation and show results of numerical experiments.

Parallelizing a Modified Particle Swarm Optimizer (PSO)

Recently, a modified Particle Swarm Optimizer (MLPSO) has been succeeded in solving truss topological optimization problems and competitive results are obtained. Since this optimizer belongs to evolutionary algorithm and plagued by high computational cost as measured by execution time, in order to reduce its execution time for solving large complex optimization problem, a parallel version for this optimizer is studied in this paper. This paper first gives an overview of PSO algorithm as well as the modified PSO, and then a design and an implementation of parallel PSO is proposed. The performance of the proposed algorithm is tested by two examples and promising speed-up rate is obtained. Final part is conclusion and outlook.

Generalized Benders’ Decomposition for topology optimization problems

This article considers the non-linear mixed 0---1 optimization problems that appear in topology optimization of load carrying structures. The main objective is to present a Generalized Benders' Decomposition (GBD) method for solving single and multiple load minimum compliance (maximum stiffness) problems with discrete design variables to global optimality. We present the theoretical aspects of the method, including a proof of finite convergence and conditions for obtaining global optimal solutions. The method is also linked to, and compared with, an Outer-Approximation approach and a mixed 0---1 semi definite programming formulation of the considered problem. Several ways to accelerate the method are suggested and an implementation is described. Finally, a set of truss topology optimization problems are numerically solved to global optimality.

A mixed integer programming for robust truss topology optimization with stress constraints

AbstractThis paper presents a mixed integer programming (MIP) formulation for robust topology optimization of trusses subjected to the stress constraints under the uncertain load. A design‐dependent uncertainty model of the external load is proposed for dealing with the variation of truss topology in the course of optimization. For a truss with the discrete member cross‐sectional areas, it is shown that the robust topology optimization problem can be reduced to an MIP problem, which is solved globally. Numerical examples illustrate that the robust optimal topology of a truss depends on the magnitude of uncertainty. Copyright © 2010 John Wiley & Sons, Ltd.

Exploiting special structure in semidefinite programming: A survey of theory and applications

Abstract Semidefinite programming (SDP) may be seen as a generalization of linear programming (LP). In particular, one may extend interior point algorithms for LP to SDP, but it has proven much more difficult to exploit structure in the SDP data during computation. We survey three types of special structures in SDP data: 1. A common ‘chordal’ sparsity pattern of all the data matrices. This structure arises in applications in graph theory, and may also be used to deal with more general sparsity patterns in a heuristic way. 2. Low rank of all the data matrices. This structure is common in SDP relaxations of combinatorial optimization problems, and SDP approximations of polynomial optimization problems. 3. The situation where the data matrices are invariant under the action of a permutation group, or, more generally, where the data matrices belong to a low dimensional matrix algebra. Such problems arise in truss topology optimization, particle physics, coding theory, computational geometry, and graph theory. We will give an overview of existing techniques to exploit these structures in the data. Most of the paper will be devoted to the third situation, since it has received the least attention in the literature so far.

Reduction of truss topology optimization

A reduction of truss topology design problem formulated by semidefinite optimization (SDO) is considered. The finite groups and their representations are introduced to reduce the stiffness and mass matrices of truss in size. Numerical results are given for both the original problem and the reduced problem to make a comparison.

TOPOLOGY OPTIMIZATION OF TRUSSES CONSISTING OF TRADITIONAL LAYOUTS

A linear Mixed Integer Programming (MIP) approach is presented for topology optimization of trusses with discrete variables. The problem with stress constraints is reformulated to linear MIP problem. The objective function is the total structural volume. It is shown in the numerical examples that an optimal combination of the traditional layouts of plane and space trusses can be successfully found by solving linear 0-1 MIP problems. A systematic approach is developed for generating the constraints on the 0-1 variables for selection of members representing the traditional layouts. The effectiveness of the proposed method is demonstrated in the numerical examples.

Mixed genetic algorithm for optimization of trusses

In this paper two strategies of optimization are compared: sequential and synchronous topology and shape optimization of trusses. Genetic algorithms are used for optimization. A task of optimization of truss withtwelve possible nodes is solved. Finite elements method is used to calculate an objective function value. Software used in calculations was created by the author.

Exploiting group symmetry in truss topology optimization

We consider semidefinite programming (SDP) formulations of certain truss topology optimization problems, where a lower bound is imposed on the fundamental frequency of vibration of the truss structure. These SDP formulations were introduced in Ohsaki et al. (Comp. Meth. Appl. Mech. Eng. 180:203–217, 1999). We show how one may automatically obtain symmetric designs, by eliminating the ‘redundant’ symmetry in the SDP problem formulation. This has the advantage that the original SDP problem is substantially reduced in size for trusses with large symmetry groups.

Topology optimization of trusses by growing ground structure method

A new method called the growing ground structure method is proposed for truss topology optimization, which effectively expands or reduces the ground structure by iteratively adding or removing bars and nodes. The method uses five growth strategies, which are based on mechanical properties, to determine the bars and nodes to be added or removed. Hence, the method can optimize the initial ground structures such that the modified, or grown, ground structures can generate the optimal solution for the given set of nodes. The structural data of trusses are manipulated using C++ standard template library and the Boost Graph Library, which help alleviate the programming efforts for implementing the method. Three kinds of topology optimization problems are considered. The first problem is a compliance minimization problem with cross-sectional areas as variables. The second problem is a minimum compliance problem with the nodal coordinates also as variables. The third problem is a minimum volume problem with stress constraints under multiple load cases. Six numerical examples corresponding to these three problems are solved to demonstrate the performance of the proposed method.