Image-based truss recognition for density-based topology optimization approach

Structural and Multidisciplinary Optimization - Topology optimization is a tool that supports the creativity of structural-designers and is used in various industries, from automotive to...     

On simultaneous optimization of truss geometry and topology

The paper addresses the classical problem of optimal truss design where cross-sectional areas and the positions of joints are simultaneously optimized. Se-veral approaches are discussed from a general point of view. In particular, we focus on the difference between simultaneous and alternating optimization of geometry and topology. We recall a rigorously mathematical approach based on the implicit programming technique which considers the classical single load minimum compliance problem subject to a volume constraint. This approach is refined leading to three new problem formulations which can be treated by methods of Mathematical Programming. In particular, these formulations cover the effect of melting end nodes, i.e., vanishing potential bars due to changes in the geometry. In one of these new problem formulations, the objective function is a polynomial of degree three and the constraints are bilinear or just sign constraints. Because heuristics is avoided, certain optimality properties can be proven for resulting structures. The paper closes with two numerical test examples.     

A note on stress-constrained truss topology optimization

The purpose of this brief note is to demonstrate that general-purpose optimization methods and codes should not be discarded when dealing with stress-constrained truss topology optimization. By using a disaggregated formulation of the considered problem, such methods may find also “singular optima”, without using perturbation techniques like the ε-relaxed approach. Received February 19, 2002     

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 Approximation (GATA) was proposed in a previous work, and showed good computational efficiency with less structural analyses. However, this optimization method easily converges to sub-optimum points, resulting in a poor ability to search for a global optimum. Therefore, to address this problem, we propose an Improved GA with a Two-level Approximation (IGATA) which includes several modifications to the approximation function and simple GA developed previously. A Branched Multi-point Approximation (BMA) function, which is efficient and without singularity, is introduced to construct a first-level approximation problem. A modified Lemonge penalty function is adopted for the fitness calculation, while an Elite Selection Strategy (ESS) is proposed to improve the quality of the initial points. The results of numerical examples confirm the lower computational cost of the algorithm incorporating these modifications. Numerous numerical experiments show good reliability of the IGATA given appropriate values for the considered parameters.     

Structural topology optimization with design-dependent pressure loads

One way to solve topology optimization of continuum structures under design-dependent pressure loads is to recover the loading surface at each step of the minimization process. During the topology evolution, the intermediate topologies obtained by using the SIMP (Solid Isotropic Material with Penalization) method actually can be regarded as gray scale images, for which the paper proposes a new material boundary identification scheme based on image segmentation technique. The Distance Regularized Level Set Evolution (DRLSE) method proposed by Li et al., IEEE Trans Image Process 19(12):3243–3254 (2010) is utilized to detect the image edge. Then the pressure boundary is represented as the zero level contour of a level set function (LSF). Inheriting the merits of the level set method, the current scheme can handle the topological change of the pressure boundary efficiently and be easily extended to the three-dimensional problems. In addition, the scheme is more stable compared with the conventional loading surface searching methods since it works well for the intermediate topologies with local scattered densities. A new optimization framework is also proposed to avoid the load sensitivity analysis. Four numerical examples are presented to show the validity and advantages of the proposed scheme.     

Two approaches for truss topology optimization: a comparison for practical use

This paper discusses ground structure approaches for topology optimization of trusses. These topology optimization methods select an optimal subset of bars from the set of all possible bars defined on a discrete grid. The objectives used are based either on minimum compliance or on minimum volume. Advantages and disadvantages are discussed and it is shown that constraints exist where the formulations become equivalent. The incorporation of stability constraints (buckling) into topology design is important. The influence of buckling on the optimal layout is demonstrated by a bridge design example. A second example shows the applicability of truss topology optimization to a real engineering stiffened membrane problem.     

A ground-structure-based representation with an element-removal algorithm for truss topology optimization

In this study, a new ground-structure-based representation for truss topology optimization is proposed. The proposed representation employs an algorithm that removes unwanted elements from trusses to obtain the final trusses. These unwanted elements include kinematically unstable elements and useless zero-force elements. Since the element-removal algorithm is used in the translation of representation codes into corresponding trusses, this results in more representation codes in the search space that are mapped into kinematically stable and efficient trusses. Since more representation codes in the search space represent stable and efficient trusses, the strategy increases meaningful competition among representation codes. This remapping strategy alleviates the problem of having large search spaces using ground structures, and encourages faster convergences. To test the effectiveness of the proposed representation, it is used with a simple multi-population particle swarm optimization algorithm to solve several truss topology optimization problems. It is found that the proposed representation can significantly improve the performance of the optimization process.     

Evolutionary truss topology optimization using a graph-based parameterization concept

A novel parameterization concept for the optimization of truss structures by means of evolutionary algorithms is presented. The main idea is to represent truss structures as mathematical graphs and directly apply genetic operators, i.e., mutation and crossover, on them. For this purpose, new genetic graph operators are introduced, which are combined with graph algorithms, e.g., Cuthill–McKee reordering, to raise their efficiency. This parameterization concept allows for the concurrent optimization of topology, geometry, and sizing of the truss structures. Furthermore, it is absolutely independent from any kind of ground structure normally reducing the number of possible topologies and sometimes preventing innovative design solutions. A further advantage of this parameterization concept compared to traditional encoding of evolutionary algorithms is the possibility of handling individuals of variable size. Finally, the effectiveness of the concept is demonstrated by examining three numerical examples.     

Growth method for size,topology, and geometry optimization of truss structures

The problem of optimally designing the topology of plane trusses has, in most cases, been dealt with as a size problem in which members are eliminated when their size tends to zero. This article presents a novel growth method for the optimal design in a sequential manner of size, geometry, and topology of plane trusses without the need of a ground structure. The method has been applied to single load case problems with stress and size constraints. It works sequentially by adding new joints and members optimally, requiring five basic steps: (1) domain specification, (2) topology and size optimization, (3) geometry optimization, (4) optimality verification, and (5) topology growth. To demonstrate the proposed growth method, three examples were carried out: Michell cantilever, Messerschmidt–Bölkow–Blohm beam, and Michell cantilever with fixed circular boundary. The results obtained with the proposed growth method agree perfectly with the analytical solutions. A Windows XP program, which demonstrates the method, can be downloaded from http://www.upct.es/~deyc/software/tto/.     

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.     

Multiobjective topology optimization of truss structures with kinematic stability repair

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.     

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.     

Exact analytical solutions for some popular benchmark problems in topology optimization

The aim of this paper is to provide the exact analytical truss solutions for some benchmark problems, which are often used as test examples in both discretized layout optimization of trusses and variable topology (or generalized) shape optimization of perforated plates under plane stress.     

Structural topology optimization with constraints on multi-fastener joint loads

This paper addresses an important problem of design constraints on fastener joint loads that are well recognized in the design of assembled aircraft structures. To avoid the failure of fastener joints, standard topology optimization is extended not only to minimize the structural compliance but also to control shear loads intensities over fasteners. It is shown that the underlying design scheme is to ameliorate the stiffness distribution over the structure in accordance with the control of load distributions over fastener joints. Typical examples are studied by means of topology optimization with joint load constraints and the standard compliance design. The effects of joint load constraints are highlighted by comparing numerical optimization results obtained by both methods. Meanwhile, resin models of optimized designs are fabricated by rapid prototyping process for loading test experiments to make sure the effectiveness of the proposed method.     

Optimization methods for truss geometry and topology design

Truss topology design for minimum external work (compliance) can be expressed in a number of equivalent potential or complementary energy problem formulations in terms of member forces, displacements and bar areas. Using duality principles and non-smooth analysis we show how displacements only as well as stresses only formulations can be obtained and discuss the implications these formulations have for the construction and implementation of efficient algorithms for large-scale truss topology design. The analysis covers min-max and weighted average multiple load designs with external as well as self-weight loads and extends to the topology design of reinforcement and the topology design of variable thickness sheets and sandwich plates. On the basis of topology design as an inner problem in a hierarchical procedure, the combined geometry and topology design of truss structures is also considered. Numerical results and illustrative examples are 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.     

Multi-objective topology and sizing optimization of truss structures based on adaptive multi-island search strategy

This paper uses genetic algorithm to handle the topology and sizing optimization of truss structures, in which a sparse node matrix encoding approach is used and individual identification technique is employed to avoid duplicate structural analysis to save computation time. It is observed that NSGA-II could not improve the convergence of non-dominated front at latter generations when solving multi-objective topology and sizing optimization of truss structures. Therefore, an adaptive multi-island search strategy for multi-objective optimization problem (AMISS-MOP) is developed to enhance the convergence. Meanwhile, an elitist strategy based on archive set is introduced to reduce the size of non-dominated sorting to improve computation efficiency. Two numeric examples are presented to demonstrate the performance of AMISS-MOP. Results show that the global Pareto front could be found by AMISS-MOP, the convergence is improved as generation increases, and the time spent on non-dominated sorting is reduced.     

A new solution for topology optimization problems with multiple loads:The guide-weight method

The guide-weight method is introduced to solve two kinds of topology optimization problems with multiple loads in this paper.The guide-weight method and its Lagrange multipliers' solution methods are presented first,and the Lagrange multipliers' soution method of problems with multiple constraints is improved by the dual method.Then the iterative formulas of the guide-weight method for topology optimization problems of minimum compliance and minimum weight are derived and coresponding numerical examples are...