Correction: MILP-based discrete sizing and topology optimization of truss structures: new formulation and benchmarking

Discrete sizing and topology optimization of truss structures subject to stress and displacement constraints has been formulated as a Mixed-Integer Linear Programming (MILP) problem. The computation time to solve a MILP problem to global optimality via a branch-and-cut solver highly depends on the problem size, the choice of design variables, and the quality of optimization constraint formulations. This paper presents a new formulation for discrete sizing and topology optimization of truss structures, which is benchmarked against two well-known existing formulations. Benchmarking is carried out through case studies to evaluate the influence of the number of structural members, candidate cross sections, load cases, and design constraints (e.g., stress and displacement limits) on computational performance. Results show that one of the existing formulations performs significantly worse than all other formulations. In most cases, the new formulation proposed in this work performs best to obtain near-optimal solutions and verify global optimality in the shortest computation time.

A new fuzzy strategy for size and topology optimization of truss structures

This paper addresses single and multiobjective topology optimization of truss-like structures using genetic algorithms (GA's). In order to improve the performance of the GA's (despite the presence of binary topology variables) a novel approach based on kinematic stability repair (KSR) is proposed. The methodology consists of two parts, namely the creation of a number of kinematically stable individuals in the initial population (IP) and a chromosome repair procedure. The proposed method is developed for both 2D and 3D structures and is shown to produce (in the single-objective case) results which are better than, or equal to, those found in the literature, while significantly increasing the rate of convergence of the algorithm. In the multiobjective case, the proposed modifications produce superior results compared to the unmodified GA. Finally the algorithm is successfully applied to a cantilevered 3D structure.

Truss structures are widely employed in the industrialized world. They appear as bridges, towers, roof supports, building exoskeletons or high technology light space structures. This paper investigates the simultaneous size, geometry and topology optimization of real life large truss structures using Genetic Algorithms (GAs) as optimizer and Finite Element Method as analyzer. In general the large truss structures are constructed from the duplication of some basic modules called bays. Thus, the final optimum design may be reached by optimizing the characteristics of the basic bays instead of optimizing the whole structure. Both single and multi-objective functions based on the mass of the structure and the maximum nodal displacement, have been considered as the cost functions. In order to have realistic optimal designs, the cross-sectional areas have been extracted from the standard profiles according to AISC codes and practical conditions are imposed to the bays. The design optimization problem is also constrained by the maximum stress, maximum slenderness ratio and the maximum and minimum cross-sectional area of the truss members. To accommodate all these constraints, two different penalty functions are proposed. The first penalty function considers the normalization of violated constraints with respect to the allowable stress or slenderness ratio. The second penalty function is a constant function, which is used to penalize the violations of the slenderness ratio. Two illustrative examples of realistic planar and space truss structures have been optimized to demonstrate the effectiveness of the proposed methodology.

Discrete ground structure topology optimization design methods have to date received considerable attention in structural engineering. An alternative class of structural topology optimization methods, which have not yet received much attention in structural engineering, but which have undergone considerable development in the past decade, are the so-called continuum formulations. In this work, a continuum structural topology optimization formulation is presented and applied to the concept design optimization of structural bracing systems that are needed to stiffen tall structures against sidesway under lateral-wind and seismic-type loading. Although demonstrated here in the context of these specific design examples, continuum structural topology optimization methods are believed to hold potential as a design tool for a wide range of civil engineering type structures. A variety of continuum topology design formulations, including static compliance minimization and eigenvalue optimization, are explored, and solution parameters are varied to show that a number of design possibilities can be realized as solutions.

Topology optimization of truss structures is considered in this paper. Trusses are widely used in various constructions: bridges, towers, roof supporting structures. Topology optimization of trusses requires large amount of computing resources. Therefore distributed computer networks are used to solve this kind of problems. In this paper a distributed branch-and-bound combinatorial algorithm for topology optimization of trusses is investigated in computational grid infrastructure and clusters of personal computers.

Structural topology constraints in topology optimization are an important research topic. The structural topology is characterized by the topological invariance of the number of holes. The holes of a structure in 3D space can be classified as internally enclosed holes and external through-holes (or tunnels). The genus is the number of tunnels. This article proposes the quotient set design variable method (QSDV) to implement the inequality constraint on the maximum genus allowed in an optimized structure for 3D structural topology optimization. The principle of the QSDV is to classify the changing design variables according to the connectivity of the elements in a structure to obtain the quotient set and update the corresponding elements in the quotient set to meet the topological constraint. Based on the standard relaxation algorithm discrete variable topology optimization method (DVTOCRA), the effectiveness of the QSDV is illustrated in numerical examples of a 3D cantilever beam.

Explicit control of 2D and 3D structural complexity by discrete variable topology optimization method

The topology of a structure is defined by its genus or number of handles. When the topology of a structure is optimized, its topology might be changed if the material state of a design cell is switched from solid to void or vice versa. In discrete topology optimization, each design cell is either solid or void and there is no topology uncertainty from any grey design cell. Point connection might cause topology uncertainty and is eradicated when hybrid discretization model is used for discrete topology optimization. However, the topology solution of an optimized structure might be uncertain when its design domain is discretized differently, which is commonly called mesh dependence problem. In this paper, the degree of genus based topology optimization strategy is introduced to circumvent this topology uncertainty. With this strategy, the genus of an optimized structure is constrained during its topology optimization process. There is no topology uncertainty even if different design domain discretizations are used. The introduced strategy is used for discrete topology optimization of structures that have multiple loading points in this paper. The presented discrete topology optimization procedure is demonstrated by examples with different degrees of genus and loading conditions.

ABSTRACTThis article presents an enhanced particle swarm optimization (EPSO) algorithm for size and shape optimization of truss structures. The proposed EPSO introduces a particle categorization mechanism into the particle swarm optimization (PSO) to eliminate unnecessary structural analyses during the optimization process and improve the computational efficiency of the PSO-based structural optimization. The numerical investigation, including three benchmark truss optimization problems, examines the efficiency of the EPSO. The results demonstrate that the particle categorization mechanism greatly reduces the computational requirements of the PSO-based approaches while maintaining the original search capability of the algorithms in solving optimization problems with computationally cheap objective function and expensive constraints.

An application of relative difference quotient algorithm to topology optimization of truss structures with discrete variables

In this paper, a technology enabling to optimize the topology of truss or frame structures with genetic algo-rithms is presented. The objective function is total mass of the structure elements. Constrains consist of equilibrium equations and conditions on stresses in the elements: stresses cannot exceed allowable value and must be above some threshold value. The values of the objective function are calculated with package ANSYS. Genetic algorithm with interface is implemented in C++. Exchange of data between ANSYS and C++ goes through temporary files. Numerical examples are presented.

In the discrete topology optimization, material state is either solid or void and there is no topology uncertainty caused by any intermediate material state. In this paper, the improved hybrid discretization model is introduced for the discrete topology optimization of structures. The design domain is discretized into quadrilateral design cells and each quadrilateral design cell is further subdivided into triangular analysis cells. The dangling and redundant solid design cells are completely eliminated from topology solutions in the improved hybrid discretization model to avoid sharp protrusions. The local stress constraint is directly imposed on each triangular analysis cell to make the designed structure safe. The binary bit-array genetic algorithm is used to search for the optimal topology to circumvent the geometrical bias against the vertical design cells. The presented discrete topology optimization procedure is illustrated by two topology optimization examples of structures.

During the last 20 years, structural optimization has become one of the most important topics of engineering applications. Design optimization of structure has been an interesting area of research in the field of engineering design for its ability to short the design cycle and to enhance product quality. Significant research activity has occurred in the area of structural optimization in the last decade. Especially for topology optimization of structure, many new theoretical, algorithmic, and computational contributions have resulted by researchers and engineers. Topology optimization is a powerful tool for global and multi-scale design of macrostructures, microstructures, and the cell of prescribed composite materials. The population based evolutionary algorithms have emerged as powerful mechanism for finding optimum solutions of complex optimization problems in engineering during the last two decades. Evolutionary computation is the study of computational systems which use ideas and get inspiration from natural evolution and adaptation [1]. The thinking has wide application in various engineering fields, such as computer science, artificial intelligence, operations research. Genetic algorithm is another kind of bio-inspired optimization method and it is playing an increasingly important role in studies of complex adaptive systems. Its application ranges from adaptive agents in economic theory to the use of machine learning techniques in the design of complex devices and structures, such as aircraft turbines and integrated circuits [2]. Optimization of structures can be classified into three categories: sizing, shaping, and topology optimization. In the topology optimization, it is concerned with the structure members and connectivity between members. In general, it is easily represented by discrete variables rather than by those used for continuous optimization problems. Topology optimization is the most difficult and complex among three categories and it is special useful in developing innovative conceptual designs. Structural optimization, in particular the topology optimization, has been identified as one of the most challenging tasks in structural design. Various techniques and approaches have been established during the last two decades. Topology optimization usually referred to as layout optimization or general shape optimization [3]. It lets engineers get the optimal topology of structure or new configurations during product design phase, as they are implementing the design of the size

This article presents a discrete variable topology optimization method to solve the simplified convective heat transfer (SCHT) design optimization modeled by Newton's law of cooling. The discrete variable topology optimization is based on the proposed sequential approximate integer programming with trust‐region. Due to the discrete variables, identifying the convective boundary, and implementing this design‐dependent convective boundary condition can be precisely undertaken. As a result, the consistent precise temperature field compared with the commercial software is captured. Besides, the interpolation scheme of the convective coefficient is unnecessary to analyze this SCHT problem. Furthermore, an analytical sensitivity formulation that can simultaneously incorporate the conductive and convective effect is also deduced. Finally, several 2D and 3D valid thermal designs are presented to illustrate the effectiveness of the method. Based on the optimized designs, we find that favorable configurations for a simplified convective problem may be hollowed structure or the dense needle‐like structure. Further, the checkerboard pattern should be interpreted as a convection oscillatory feature but not the discretization error because it cannot be eliminated by using higher‐order elements.