Efficient size and shape optimization of truss structures subject to stress and local buckling constraints using sequential linear programming

The advance in digital fabrication technologies and additive manufacturing allows for the fabrication of complex truss structure designs but at the same time posing challenging structural optimization problems to capitalize on this new design freedom. In response to this, an iterative approach in which Sequential Linear Programming (SLP) is used to simultaneously solve a size and shape optimization sub-problem subject to local stress and Euler buckling constraints is proposed in this work. To accomplish this, a first order Taylor expansion for the nodal movement and the buckling constraint is derived to conform to the SLP problem formulation. At each iteration a post-processing step is initiated to map a design vector to the exact buckling constraint boundary in order to facilitate the overall efficiency. The method is verified against an exact non-linear optimization problem formulation on a range of benchmark examples obtained from the literature. The results show that the proposed method produces optimized designs that are either close or identical to the solutions obtained by the non-linear problem formulation while significantly decreasing the computational time. This enables more efficient size and shape optimization of truss structures considering practical engineering constraints.     

Randomized line search techniques in combined GA for discrete sizing optimization of truss structures

This paper presents two randomized line search techniques, each combined with a genetic algorithm (GA), to improve the convergence and the accuracy ratio for discrete sizing optimization of truss structures. The first technique is a simple one-dimensional line search in which design variable axes are selected randomly as search directions. The second is a line search technique whose search direction is determined randomly by fitness function values and differences in the genotypes of individuals. To apply the above-mentioned line search techniques without difficulty, real coding is adopted for discrete problems. The line search techniques are applied to discrete optimization problems of minimum-weight truss structures subjected to stress and displacement constraints. The proposed techniques provide convergence to better solutions than a conventional GA.     

Optimum design of truss topology under buckling constraints

The present paper studies the optimum design of truss topology under buckling constraints based on a new formulation of the problem. Through the incorporation of a global system stability constraint into the problem formulation, isolated compressive bars are eliminated from the final optimal topology. Furthermore, by including overlapping bars in the initial ground structure, the difficulty caused by hinge cancellation as pointed out by Rozvany (1996) can be overcome. Also, the importance of inclusion of compatibility conditions in the problem formulation is demonstrated. Finally, several numerical examples are presented for demonstration of the effectiveness of the proposed approach.     

Optimal drug dosage subject to constraints

Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 174–176, November–December, 1993.     

Enhanced hybrid metaheuristic algorithms for optimal sizing of steel truss structures with numerous discrete variables

The advent of modern computing technologies paved the way for development of numerous efficient structural design optimization tools in the recent decades. In the present study sizing optimization problem of steel truss structures having numerous discrete variables is tackled using combined forms of recently proposed metaheuristic techniques. Three guided, and three guided hybrid metaheuristic algorithms are developed by integrating a design oriented strategy to the stochastic search properties of three recently proposed metaheuristic optimization techniques, namely adaptive dimensional search, modified big bang-big crunch, and exponential big bang-big crunch algorithms. The performances of the proposed guided, and guided hybrid metaheuristic algorithms are compared to those of standard variants through optimum design of real-size steel truss structures with up to 728 design variables according to AISC-LRFD specification. The numerical results reveal that the hybrid form of adaptive dimensional search and exponential big bang-big crunch algorithm is the most promising algorithm amongst the other investigated techniques.     

Hypotrochoid spiral optimization approach for sizing and layout optimization of truss structures with multiple frequency constraints

Engineering with Computers - The primary aim of this article is to present a new improved version of the spiral optimization algorithm (SPO) for shape and size optimization of truss structures...     

Optimal truss design including plastic collapse constraints

A new algorithm is presented for size optimization of truss structures with any kind of smooth objectives and constraints, together with constraints on the collapse loading obtained by limit analysis, for several loading conditions. The main difficulty of this problem is the fact that the collapse loading is a nonsmooth function of the design variables. In this paper we avoid nonsmooth optimization techniques based on the fact that limit analysis constraints are linear by parts. Our approach is based on a feasible directions interior point algorithm for nonlinear constrained optimization. Three illustrative examples are discussed. The numerical results show that the calculation effort when limit analysis constraints are included is only slightly increased with respect to classic constraints.     

Optimal truss sizing based on explicit Taylor series expansions

This paper deals with the classical structural optimization problem of minimizing the weight of a truss structure, subject to constraints on displacements and stresses. Design variables are the cross-section areas of the elements. The main theoretical result presented is that the complete Taylor series expansions of all constraint functions can be explicitly expressed, once the displacements corresponding ton specific load cases have been calculated, wheren = the number of elements. Based on this theoretical result, new explicit approximations of the constraint functions are suggested. These approximations are shown to have some interesting theoretical properties. They also seem to work very well in practice, when included in an optimization scheme.     

A new approximation of the constraints in truss sizing problems: An explicit second order approximation which is exact for statically determinate truss structures

A new explicit approximation of the constraint functions (displacements and stresses) in truss sizing problems is presented and shown to have the following properties:

—it is a second order approximation;

—it is exact if at most two variables (cross-sectional areas) are varied simultaneously;

—it is exact when applied on statically determinate structures.

The approximation is included in an optimization scheme for truss sizing and applied on several test problems. The optimal solution is always obtained, with a very high accuracy, after less than 5 global iterations, even on problems where traditional first order methods require up to 40 global iterations.     

Difficulties in truss topology optimization with stress and local buckling constraints

The aim of this note is to discuss problems associated with local buckling constraints in the context of topology optimization. It is shown that serious difficulties are encountered unless additional measures are introduced.     

Topological optimization of structures subject to Von Mises stress constraints

The topological asymptotic analysis provides the sensitivity of a given shape functional with respect to an infinitesimal domain perturbation. Therefore, this sensitivity can be naturally used as a descent direction in a structural topology design problem. However, according to the literature concerning the topological derivative, only the classical approach based on flexibility minimization for a given amount of material, without control on the stress level supported by the structural device, has been considered. In this paper, therefore, we introduce a class of penalty functionals that mimic a pointwise constraint on the Von Mises stress field. The associated topological derivative is obtained for plane stress linear elasticity. Only the formal asymptotic expansion procedure is presented, but full justifications can be deduced from existing works. Then, a topology optimization algorithm based on these concepts is proposed, that allows for treating local stress criteria. Finally, this feature is shown through some numerical examples.     

Optimal design of truss structures using parallel computing

Parallel design optimization of large structural systems calls for a multilevel approach to the optimization problem. The general optimization problem is decomposed into a number of non-interacting suboptimization problems on the first level. They are controlled from the second level through coordination variables. Thus, the solutions of the independent first-level subsystems are directed towards the overall system optimum. In the present paper, optimal design of truss structures using parallel computing technique is described. In this method, optimization of a large truss structure has been carried out by decomposing the structure into sub-domains and suboptimization tasks. Each sub-domain has independent design variables and a small number of behaviour constraints. The two-level sub-domain optimum design approach is summarized by several numerical examples with speedups and efficiencies of algorithms on message passing systems. It has been noticed that the efficiency of the algorithm for design optimization increases with the size of the structure.     

Optimal design of truss structures using ant algorithm

An ant algorithm, consisting of the Ant System and API (after “apicalis” in Pachycondyla apicalis) algorithms, was proposed in this study to find optimal truss structures to achieve minimum weight objective under stress, deflection, and kinematic stability constraints. A two-stage approach was adopted in this study; first, the topology of the truss structure was optimized from a given ground structure employing the Ant System algorithm due to its discrete characteristic, and then the size and/or shape of member was optimized utilizing the API algorithm. The effectiveness of the proposed ant algorithm was evaluated through numerous different 2-D and 3-D truss-structure problems. The proposed algorithm was observed to find truss structures better than those reported in the literature. Moreover, multiple different truss topologies with almost equal overall weights can be found simultaneously.     

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.     

Optimal design of truss structures for size and shape with frequency constraints using a collaborative optimization strategy

A new metaheuristic strategy is proposed for size and shape optimization problems with frequency constraints. These optimization problems are considered to be highly non-linear and non-convex. The proposed strategy extends the idea of using a single optimization process to a series of collaborative optimization processes. In this study, a modified teaching-learning-based optimization (TLBO), which is a relatively simple algorithm with no intrinsic parameters controlling its performance, is utilized in a collaborative framework and introduced as a higher-level TLBO algorithm called school-based optimization (SBO). SBO considers a school with multiple independent classrooms and multiple teachers with inter-classroom collaboration where teachers are reassigned to classrooms based on their fitness. SBO significantly improves the both exploration and exploitation capabilities of TLBO without increasing the algorithm's complexity. In addition, since the SBO algorithm uses multiple independent classrooms with interchanging teachers, the algorithm is less likely to be influenced by local optima. A parametric study is conducted to investigate the effects of the number of classes and the class size, which are the only parameters of SBO. The SBO algorithm is applied to five benchmark truss optimization problems with frequency constraints and the statistical results are compared to other optimization techniques in the literature. The quality and robustness of the results indicate the efficiency of the proposed SBO algorithm.     

最大误差约束下最小二乘准则FIR滤波器设计

考虑了一类加权最小二乘复系数FIR滤波器的设计问题,要求滤波器通带及止带内频点的频率响应与期望响应误差小于给定的最大误差门限.结合半正定优化理论,经过数学推导,将该设计问题转换为半正定优化问题,然后借助内点方法给出最优滤波器系数.另外,该方法能够检测出给定约束条件的优化问题是否存在可行解,从而可以使设计者能够以自适应的方式调整约束条件.仿真结果表明设计算法是十分有效的,并具有很好的稳健性.     

Discrete optimization of geometrically nonlinear truss structures under stability constraints

The paper deals with discrete optimization of elastic trusses with geometrical nonlinear behaviour and constraints on stability. The problem consists of minimizing the weight and determining the optimal member distribution so that the external load does not cause a loss of stability of the structure. Member cross-sections are selected from a catalogue of available sections. Element stresses, elment stability and global structural stability constraints are considered. A controlled enumeration method according to the increasing value of the objective function is applied. Shallow space trusses are numerically analysed.     

Design optimization and sensitivity analysis on buckling stability of piezoelectric truss structures

The design optimization of buckling behavior is studied for piezoelectric intelligent truss structures. First, on the basis of mechanical–electric coupling equation and considering electric load and mechanical loads together, the finite element model of piezoelectric trusses has been built up. Then, the computational formula has been derived for the design sensitivities of critical buckling load factor of the structure with respect to size and shape design variables. The electric voltage is taken as a new kind of design variable and the calculation method of critical load buckling factor with respect to the electric voltage variables is proposed. Particularly, the variations of the loads and pre-buckling inner forces with design variables have been accounted for. Finally, the sequential linear programming algorithm is employed to solve the optimization problem, and a new method of controlling structural buckling stability by optimizing the voltages of piezoelectric active bars is proposed. Numerical examples given in the paper have demonstrated the effectiveness of the methods presented.     

Difficulties in truss topology optimization with stress,local buckling and system stability constraints

A serlous difficulty in topology optimization with only stress andlocal buckling constraints was pointed out recently by Zhou (1996a). Possibilities for avoiding this pitfall are (i) inclusion of system stability constraints and (ii) application of imperfections in the ground structure. However, it is shown in this study that the above modified procedures may also lead to erroneous solutions which cannot be avoided without changing the ground structure.