Truss Topology Optimization Research Articles

Colliding bodies optimization for size and topology optimization of truss structures

This paper presents the application of a recently developed meta-heuristic algorithm, called Colliding Bodies Optimization (CBO), for size and topology optimization of steel trusses. This method is based on the one-dimensional collisions between two bodies, where each agent solution is considered as a body. The performance of the proposed algorithm is investigated through four benchmark trusses for minimum weight with static and dynamic constraints. A comparison of the numerical results of the CBO with those of other available algorithms indicates that the proposed technique is capable of locating promising solutions using lesser or identical computational effort, with no need for internal parameter tuning.

Solving truss topological optimization via Swarm Intelligence

Structural topological optimization is the most general form of structural optimization and requires a less detailed description of the concept. One of the most exciting and challenging problems in this field is to find optimized layouts with minimization of compliance (maximization of stiffness) for a given total mass of the structure discretized by truss members, which cannot be well solved by evolutionary algorithms. Particle Swarm Optimization (PSO) is a new paradigm of Swarm Intelligence which is inspired by concepts from ‘Social Psychology' and ‘Artificial Life'. PSO is particularly a preferable candidate to solve highly nonlinear, non-convex and even discontinuous problems and has been applied to many different kinds of optimization problems. The motivation of this paper is to propose an enhanced Lbest based PSO and geometrical consistency check tightly connecting to the ground structure approach to break through in this optimization field. Through a popular benchmark test, two kinds of Modified Lbest based PSO (MLPSO) exhibited competitive performance due to improved global searching ability.

Solving truss topological optimization with discrete design variables via swarm intelligence

Structural topological optimization is the most general form of structural optimization and one of the most challenging research problems in this field is to find optimized layouts with minimization of compliance (maximization of stiffness) for a given total mass of the structure discretized by truss members in terms of discrete design variables, which has not been well solved by evolutionary algorithms. Particle Swarm Optimization (PSO) is a new paradigm of Swarm Intelligence which is inspired by concepts from ‘Social Psychology’ and ‘Artificial Life’. PSO is particularly a preferable candidate to solve highly nonlinear, non-convex and even discontinuous problems and has been applied to many different kinds of optimization problems. The motivation of this paper is to use the Modified Lbest based PSO (MLPSO) and geometrical consistency check tightly connecting to the ground structure approach to break through in this kind of optimization problems. In order to handle discrete design variables, rounding-off strategy is used in this paper. Furthermore, external quadratic penalty function is chose to deal with constraints. Through a popular benchmark test, two kinds of MLPSO exhibited competitive performance due to improved global searching ability.

Convex topology optimization for hyperelastic trusses based on the ground-structure approach

Most papers in the literature, which deal with topology optimization of trusses using the ground structure approach, are constrained to linear behavior. Here we address the problem considering material nonlinear behavior. More specifically, we concentrate on hyperelastic models, such as the ones by Hencky, Saint-Venant, Neo-Hookean and Ogden. A unified approach is adopted using the total potential energy concept, i.e., the total potential is used both in the objective function of the optimization problem and also to obtain the equilibrium solution. We proof that the optimization formulation is convex provided that the specific strain energy is strictly convex. Some representative examples are given to demonstrate the features of each model. We conclude by exploring the role of nonlinearities in the overall topology design problem.

Exact Truss Topology Optimization for Multiple and Probabilistic Loads

Exact Truss Topology Optimization for Multiple and Probabilistic Loads

A semidefinite programming approach to robust truss topology optimization under uncertainty in locations of nodes

This paper addresses truss topology optimization taking into account robustness to uncertainty in the truss geometry. Specifically, the locations of nodes are assumed not to be known precisely and the compliance in the worst case is attempted to be minimized. We formulate a semidefinite programming problem that serves as a safe approximation of this robust optimization problem. That is, any feasible solution of the presented semidefinite programming problem satisfies the constraints of the original robust optimization problem. Since a semidefinite programming problem can be solved efficiently with a primal-dual interior-point method, we can find a robust truss design efficiently with the proposed semidefinite programming approach. A notable property of the proposed approach is that the obtained truss is guaranteed to be stable. Numerical experiments are performed to illustrate that the optimal truss topology depends on the magnitude of uncertainty.

Fundamentals of exact multi-load topology optimization – stress-based least-volume trusses (generalized Michell structures) – Part I: Plastic design

In this two-part paper, Michell’s century-old stress-based single-load truss topology optimization paper is extended to multiple load conditions. In Part I, so-called ‘optimal plastic design’ of multi-load trusses is reviewed, which is based on ultimate (limit or collapse) load principles and requires only statical admissibility of the solution. However, its connections to ‘optimal elastic design’ will be explained, and these will be used in Part II for the full extension of Michell’s theory to elastic multi-load problems, which will fill a significant gap in our knowledge.

Resolving issues with member buckling in truss topology optimization using a mixed variable approach

In this paper, several issues related to member buckling in truss topology optimization are treated. In the conventional formulations, where cross-sectional areas of ground structure members are the design variables, member buckling constraints are known to be very difficult to handle, both numerically and theoretically. Buckling constraints produce a feasible set that is non-connected and non-convex. Furthermore, the so-called jump in the buckling length phenomenon introduces severe difficulties for determining the correct buckling strength of parallel consecutive compression members. These issues are handled in the paper by employing a mixed variable formulation of truss topology optimization problems. In this formulation, member buckling constraints become linear. Parallel consecutive members of the ground structure are identified as chains, and overlapping members are added to the ground structure between each pair of nodes of a chain. Buckling constraints are written for every member, and linear constraints on the binary member existence variables disallow impractical topologies. In the proposed approach, Euler buckling as well as buckling according to various design codes, can be incorporated. Numerical examples demonstrate that the optimum topology depends on whether the buckling constraints are derived from Euler's theory or from design codes.

Exact compliance-based multi-load truss topology optimization revisited – also background material for an Authors’ Reply to a Discussion by Logo

The main aim of this paper is to present a general theory of exact compliance-based, multi-load truss topology optimization, including an illustrative example and its numerical verifications. A minor aim is to provide background material for an Authors' Reply to a Discussion by Logo of a paper by Rozvany and Maute.

Sizing, geometry and topology optimization of trusses using force method and supervised charged system search

In this article, the force method and Charged System Search (CSS) algorithm are used for the analysis and optimal design of truss structures. The CSS algorithm is employed as the optimization tool and the force method is utilized for analysis. In this paper in addition to member's cross sections, redundant forces, geometry and topology variables are considered as the optimization variables. Minimum complementary energy principle is used directly to analyze the structure. In the presented method, redundant forces are calculated by the CSS in order to minimize the energy function. Combination of the CSS and force method leads to an efficient algorithm in comparison to some of the optimization algorithms.

Truss topology optimization with discrete design variables by outer approximation

Several variants of an outer approximation method are proposed to solve truss topology optimization problems with discrete design variables to proven global optimality. The objective is to minimize the volume of the structure while satisfying constraints on the global stiffness of the structure under the applied loads. We extend the natural problem formulation by adding redundant force variables and force equilibrium constraints. This guarantees that the designs suggested by the relaxed master problems are capable of carrying the applied loads, a property which is generally not satisfied for classical outer approximation approaches applied to optimal design problems. A set of two- and three-dimensional benchmark problems are solved and the numerical results suggest that the proposed approaches are competitive with other special-purpose global optimization methods for the considered class of problems. Numerical comparisons indicate that the suggested outer approximation algorithms can outperform standard approaches suggested in the literature, especially on difficult problem instances.

Simultaneous Topology and Size Optimization of 2D and 3D Trusses Using Evolutionary Structural Optimization with regard to Commonly Used Topologies

One of many optimization techniques is the evolutionary structural optimization (ESO), based on the idea that an optimal structure can be achieved by gradually removing ineffectively used materials from the design domain. Production of a multistage optimization is often proposed to reach the best overall solution. In the first stage, the structure is optimized according to a topology criterion, and, in the second stage, sizing optimization is carried out. The efficiency of such an approach is questionable as a fixed topology, for the second stage optimization may not be the most favorable before sizing optimization is carried out. In this paper, the simultaneous topology and size optimization of trusses using the ESO algorithm are discussed. A number of numerical examples are presented to research capacity to achieve optimal solutions for a structural problem. The topology design of the initial design domain is based on commonly used designs for multistorey trusses constructed from straight members. Therefore, the cases are of a slender shape and made from a combination of presented internal designs. The case studies will present an evaluation to show whether the described optimization approach can be beneficial in structural design for the purpose of steel framework designs.

1211 混合整数2次錐計画を用いたトラスの離散最適化法(最適化手法・理論)

This paper presents a truss topology optimization method for finding the minimum compliance design in which only a limited number of different cross-section sizes are used. The member cross-sectional areas are considered either discrete design variables or continuous design variables. In both cases the optimization problem of interest can be reduced to a mixed-integer second-order cone programming problem. The global optimal solution is then computed by using an existing algorithm based on a branch-and-cut method.

Dynamic non-probabilistic reliability-based topology optimization of truss with uncertain-but-bounded parameters

In this paper, non-probabilistic reliability indices for frequency and static displacement constraints are analyzed based on the ellipse convex model of elastic modulus and mass density. The dynamic non-probabilistic reliability-based topology optimization model of a truss is built, where the cross-sectional areas and nodal topology variables are taken as design variables. The objective is to minimize the structural total mass. Constraints are imposed on static stresses and non-probabilistic reliability indices of static displacement and natural frequency. A genetic algorithm is used as the optimization method to find optimal solutions in the outer loop and an analysis method is adopted to seek non-probabilistic reliability index according to implicit forms of the limit state function in the inner loop. Results of numerical examples show that the optimal mass of a non-probabilistic reliability-based topology optimization is larger than that of the deterministic topology optimization and the optimal mass increases with the increase of the non-probabilistic reliability requirement in order to ensure structural safety.

Improved genetic algorithm with two-level approximation for truss topology optimization

Truss topology optimization using Genetic Algorithms (GAs) usually requires large computational cost, especially for large-scale problems. To decrease the structural analyses, a GA with a Two-level...

Redundant robust topology optimization of truss

A common problem in the optimization of structures is the handling of uncertainties in the parameters. If the parameters appear in the constraints, the uncertainties can lead to an infinite number of constraints. Usually the constraints have to be approximated by finite expressions to generate a computable problem. Here, using the example of topology optimization of a truss, a method is proposed to deal with such uncertainties by utilizing robust optimization techniques. This leads to an approach without the necessity of any approximation. Another common problem in the optimization of structures is the consideration of possible damages. The typical approach is to prevent these damages by a convenient structure—this concept is known as safe-life. The method developed here applies the principle of redundancy to resist damages, which is the design philosophy of fail-safe. In general this leads to a high dimensional partitioning problem. By using a linear ansatz we get a computable problem. Finally, robust and redundant methods are combined, and simple numerical examples of typical problems illustrate the application of the methods. Our new redundant robust topology optimization of truss, based on known concepts of different research fields, gives a structure which is not only “safe” for parameter perturbations but for failure of bars, too. This introduces the fail-safe concept to structural optimization.

Optimal Design of CNC Crushing Machine Steady Based on Genetic Algorithm

The Genetic Algorithms In engineering structure optimization design includes Truss Structure optimization, Shape and topology optimization, Composite materials optimization, layout optimization, Multi-Objective Optimization. This paper combined with engineering background , Selecting the Pressing gear of CNC crushing Machine Steady as starting point for Optimization modeling .Analyzing the simple conditions theoretical physical model of CNC Crushing Machine Steady, Reasonably selected design variables ,using Conventional Methods and genetic algorithms to optimize the Steady ,obtainning every iterative step relevant data under the two methods, and analyzing the results ,analysis the accuracy of the optimize results through the stress and displacement map .

Exact truss topology optimization for external loads and friction forces

In replying to a valuable Discussion by Mariano Vazquez Espi, the authors show that the problem of friction forces in general can be handled by the Prager-Rozvany layout theory, and the optimal Michell layout does not always correspond to the maximum value of the static friction force. Moreover, it is explained that discontinuities in the specific cost function can be accommodated by an extended version of the Prager-Shield optimality criteria, which was already demonstrated in the second author's first (1976) book.

Reliability-based topology optimization of trusses with stochastic stiffness

A new method is proposed for reliability-based topology optimization of truss structures with random geometric imperfections and material variability. Such imperfections and variability, which may result from manufacturing processes, are assumed to be small in relation to the truss dimensions and mean material properties and normally distributed. Extensive numerical evidence suggests that the trusses, when optimized in terms of a displacement-based demand metric, are characterized by randomness in the stiffness that follow the Gumbel distribution. Based on this observation, it was possible to derive analytical expressions for the structural reliability, enabling the formulation of a computationally efficient single-loop reliability-based topology optimization algorithm. Response statistics are estimated using a second-order perturbation expansion of the stiffness matrix and design sensitivities are derived so that they can be directly used by gradient-based optimizers. Several examples illustrate the accuracy of the perturbation expressions and the applicability of the method for developing optimal designs that meet target reliabilities.

Applications of truss topology optimization in the design of reinforced concrete structures using „Strut and Tie” models

Although Strut and Tie models are often used in practical design due to their apparent concept based on truss analysis, the creation of a model consistent with behaviour of the real structure is not an easy task. Frame corner model considered in the paper and presented in code [7] and article [8] exemplifies the problem. The authors proposed a method of automatic generating of ST models by making use of truss topology optimization (volume minimization problem). The method is based on classical ground structure approach. The authors introduced a method of including the cost of nodes in the objective function, which allowed to obtain solutions consisting of rationally small number of bars (unlike Michell’s structures). Moreover, algorithms ensuring consistency with Eurocode requirements were developed. The method was implemented in computer program. With the use of the software the authors proposed an alternative ST model for the frame corner, which requires considerably less reinforcement steel in comparison with the model suggested by the code. The versatility of the program was well proven in several other examples of plane stress problems in reinforced concrete design.