Structural Topology Optimization Problems Research Articles

Research on Techniques of Structural Topology Optimization Using Cellular Automaton

By integrating cellular automaton (CA) theory into topology optimization of continuum, the local rule is defined for sensitivity analysis and updating of the design variable, according to the analysis of the structural mechanical response. Topology optimization design of loaded structure is conducted using minimal compliance as the optimization objective. The optimal distribution of material in the design domain is finally obtained. Comparing to other algorithms, the local rule has proved to be computationally efficient to solve structural topology optimization problems. The resulting optimal structures are free of numerical instabilities such as the checkerboard patterns and mesh dependency.

Piecewise constant level set method for structural topology optimization with MBO type of projection

In this paper, we combine a Piecewise Constant Level Set (PCLS) method with a MBO scheme to solve a structural shape and topology optimization problem. The geometrical boundary of structure is represented implicitly by the discontinuities of PCLS functions. Compared with the classical level set method (LSM) for solving Hamilton---Jacobi partial differential equation (H-J PDE) we don't need to solve H-J PDE, thus it is free of the CFL condition and the reinitialization scheme. For solving optimization problem under some constraints, Additive Operator Splitting (AOS) and Multiplicative Operator Splitting (MOS) schemes will be used. To increase the convergency speed and the efficiency of PCLS method we combine this approach with MBO scheme. Advantages and disadvantages are discussed by solving some examples of 2D structural topology optimization problems.

A binary particle swarm optimization for continuum structural topology optimization

The particle swarm optimization (PSO) algorithm, a relatively recent bio-inspired approach to solve combinatorial optimization problems mimicking the social behavior of birds flocking, is applied to problems of continuum structural topology design for the purpose of investigating optimal topologies and automatically creating innovative solutions. An overview of the PSO and binary PSO algorithms are first described. A discretized topology design representation and the method for mapping binary particle into this representation are then detailed. Subsequently, a modified binary PSO algorithm that adopts the concept of genotype–phenotype representation is illustrated. Several well-studied examples from structural topology optimization problems of minimum weight and minimum compliance are used to demonstrate its efficiency and versatility. The results indicate the effectiveness of the proposed algorithm and its ability to find families of structural topologies.

Application of the Structural Topological Optimization in the Column Stability Design

In this paper, according to the ICM (Independent Continuous Mapping) method, the topology optimization problem of continuum structures is solved. The topology optimization model for the continuum structure is constructed, which minimized weight as the objective function and was subjected to the buckling constraints. Based on the Taylor expansion, the filtering function and the Rayleigh quotient, the objective function and the buckling constraint are approximately expressed as the explicit function. The optimization model is translated into a dual programming and solved by the sequence second-order programming. Finally, the compressed bar examples are presented. They verified the length coefficient which is converted into stability bar hinged at both ends, identified the location of bottlenecks in topological structures. According to the results, more reasonable topological structures were given.

A two phase approach based on skeleton convergence and geometric variables for topology optimization using genetic algorithm

This paper introduces a set of skeleton operators for characterizing topologies evolving in a bit-array represented structural topology optimization problem. It is shown that the design generally converges to a stable skeleton fairly early in the optimization process. It is observed that further optimization is more about finding optimal gross shape for the various branches of the converged skeleton and the bit-array representation is not appropriate. A two-phase approach to topology optimization is proposed in which the first phase, where bit-array is used to represent the topology, ends with the detection of stabilization of skeleton, and the second phase proceeds further with the geometry based representation that directly addresses gross variation in shape of the branches of the converged skeleton. Genetic Algorithm has been used for optimization in both the phases. The efficiency and effectiveness of the use of skeleton operators and geometric variables for identification of convergence in the first phase and optimization in the second phase respectively is demonstrated.

Topology optimization of structures using modified binary differential evolution

Differential evolution (DE) is an efficient population based algorithm used to solve real-valued optimization problems. It has the advantage of incorporating relatively simple and efficient mutation and crossover operators. However, the DE operator is based on floating-point representation only, and is difficult to use when solving combinatorial optimization problems. In this paper, a modified binary differential evolution (MBDE) based on a binary bit-string framework with a simple and new binary mutation mechanism is proposed. Two test functions are applied to verify the MBDE framework with the new binary mutation mechanism, and four structural topology optimization problems are used to study the performance of the proposed MBDE algorithm. The experimental studies show that the proposed MBDE algorithm is not only suitable for structural topology optimization, but also has high viability in terms of solving numerical optimization problems.

An optimization method for structural topology optimization problems with stress constraint and any initial configuration requirements

Stress related problems have not been given the same attention as the minimum compliance problem in the literature. A topological optimization method of continuum structures with the stress constraints and any initial configuration requirements has been presented in this paper. In order to greatly reduce stress sensitivity cost, all element stress constraints of the structure being optimized under a load case are replaced by the most potential active stress constraints of several design sub-domains and a generalized average stress constraint. And the whole optimization process is divided into two optimization adjustment phases and a phase transferring step operation. Firstly, an optimization model dealing with varying stress limits and design space adjustments, is built. Secondly, a procedure is proposed to solve the optimization problem of the first optimization adjustment phase. This design space adjustment capability will not affect the property of the proposed method convergence. Then, a heuristic algorithm, that is based on structural gradient information, is given to make the design structural topology be of solid/empty property during the second optimization adjustment phase. The two examples show that different initial design structure can be selected, and this method is robust and practicable.

An Enhanced Binary Particle Swarm Optimization for Structural Topology Optimization

Particle swarm optimization (PSO), a heuristic optimization method, has been successfully applied in solving many optimization problems in real-value search space. The original binary particle swarm optimization (BPSO) uses the concept of bit flipping of the binary string to convert the velocity from a real code into a binary code. However, the conversion process cannot be reversed, and it is difficult to extend this framework to solve certain discrete optimization problems. An enhanced binary particle swarm algorithm is proposed in this study based on pure binary bit-string frameworks to deal with structural topology optimization problems. Further, two enhancement strategies, stress-based strategy and pair-switched strategy, were developed to improve the performance of the proposed algorithm for topology optimization of structure. The results of experimental cases demonstrated in this study show that the proposed enhanced binary particle swarm optimization (EBPSO) with two developed strategies is an efficient population-based approach for finding the optimal design for structural topology optimization problems of minimum compliance design and minimum weight design.

Design of 2-DOF Compliant Mechanisms to Form Grip-and-Move Manipulators for 2D Workspace

This paper demonstrates the design of compliant grip-and-move manipulators by structural optimization using genetic algorithms. The manipulator is composed of two compliant mechanisms (each with two degrees of freedom) that work like two fingers so that the manipulator can grip an object and convey it from one point to another anywhere within a two-dimensional workspace. The synthesis of such compliant mechanisms is accomplished by formulating the problem as a structural topology and shape optimization problem with multiple objectives and constraints to achieve the desired behavior of the manipulator. A multiobjective genetic algorithm is then applied coupled with an enhanced morphological representation for defining and encoding the structural geometry variables. The solution framework is integrated with a nonlinear finite element code for large-displacement analyses of the compliant structures to compute the paths generated by these mechanisms, with the resulting optimal designs used to realize various manipulator configurations.

On some fundamental properties of structural topology optimization problems

We study some fundamental mathematical properties of discretized structural topology optimization problems. Either compliance is minimized with an upper bound on the volume of the structure, or volume is minimized with an upper bound on the compliance. The design variables are either continuous or 0–1. We show, by examples which can be solved by hand calculations, that the optimal solutions to the problems in general are not unique and that the discrete problems possibly have inactive volume or compliance constraints. These observations have immediate consequences on the theoretical convergence properties of penalization approaches based on material interpolation models. Furthermore, we illustrate that the optimal solutions to the considered problems in general are not symmetric even if the design domain, the external loads, and the boundary conditions are symmetric around an axis. The presented examples can be used as teaching material in graduate and undergraduate courses on structural topology optimization.

A study on X-FEM in continuum structural optimization using a level set model

In this paper, we implement the extended finite element method (X-FEM) combined with the level set method to solve structural shape and topology optimization problems. Numerical comparisons with the conventional finite element method in a fixed grid show that the X-FEM leads to more accurate results without increasing the mesh density and the degrees of freedom. Furthermore, the mesh in X-FEM is independent of the physical boundary of the design, so there is no need for remeshing during the optimization process. Numerical examples of mean compliance minimization in 2D are studied in regard to efficiency, convergence and accuracy. The results suggest that combining the X-FEM for structural analysis with the level set based boundary representation is a promising approach for continuum structural optimization.

Improvements in the treatment of stress constraints in structural topology optimization problems

Topology optimization of continuum structures is a relatively new branch of the structural optimization field. Since the basic principles were first proposed by Bendsøe and Kikuchi in 1988, most of the work has been dedicated to the so-called maximum stiffness (or minimum compliance) formulations. However, since a few years different approaches have been proposed in terms of minimum weight with stress (and/or displacement) constraints. These formulations give rise to more complex mathematical programming problems, since a large number of highly non-linear (local) constraints must be taken into account. In an attempt to reduce the computational requirements, in this paper, we propose different alternatives to consider stress constraints and some ideas about the numerical implementation of these algorithms. Finally, we present some application examples.

Structural topology optimization using ant colony optimization algorithm

The ant colony optimization (ACO) algorithm, a relatively recent bio-inspired approach to solve combinatorial optimization problems mimicking the behavior of real ant colonies, is applied to problems of continuum structural topology design. An overview of the ACO algorithm is first described. A discretized topology design representation and the method for mapping ant's trail into this representation are then detailed. Subsequently, a modified ACO algorithm with elitist ants, niche strategy and memory of multiple colonies is illustrated. Several well-studied examples from structural topology optimization problems of minimum weight and minimum compliance are used to demonstrate its efficiency and versatility. The results indicate the effectiveness of the proposed algorithm and its ability to find families of multi-modal optimal design.

Block aggregation of stress constraints in topology optimization of structures

Structural topology optimization problems have been traditionally stated and solved by means of maximum stiffness formulations. On the other hand, some effort has been devoted to stating and solving this kind of problems by means of minimum weight formulations with stress (and/or displacement) constraints. It seems clear that the latter approach is closer to the engineering point of view, but it also leads to more complicated optimization problems, since a large number of highly non-linear (local) constraints must be taken into account to limit the maximum stress (and/or displacement) at the element level. In this paper, we explore the feasibility of defining a so-called global constraint, which basic aim is to limit the maximum stress (and/or displacement) simultaneously within all the structure by means of one single inequality. Should this global constraint perform adequately, the complexity of the underlying mathematical programming problem would be drastically reduced. However, a certain weakening of the feasibility conditions is expected to occur when a large number of local constraints are lumped into one single inequality. With the aim of mitigating this undesirable collateral effect, we group the elements into blocks. Then, the local constraints corresponding to all the elements within each block can be combined to produce a single aggregated constraint per block. Finally, we compare the performance of these three approaches (local, global and block aggregated constraints) by solving several topology optimization problems.

Topology optimization of multiphase material structures under design dependent pressure loads

In this paper, design-dependent pressure loads are addressed for topology optimization problems of multiphase material structures. The modeling of unidirectional load distribution with respect to the material density is studied and mathematical formulations of exponential form are established for the first time to model unidirectional pressure loads on the moving surface and contact surface of fixed shape, respectively. In both load cases, sensitivity analysis scheme of the structural compliance is developed to show the effects of the design-dependent loads. By means of 2D and 3D numerical examples, comparative studies are performed in detail to show the effects of different load cases upon the optimal configurations. It is seen that the proposed design procedure is reliable to deal with the design-dependent load cases.

Piecewise constant level set method for structural topology optimization

AbstractIn this paper, a piecewise constant level set (PCLS) method is implemented to solve a structural shape and topology optimization problem. In the classical level set method, the geometrical boundary of the structure under optimization is represented by the zero level set of a continuous level set function, e.g. the signed distance function. Instead, in the PCLS approach the boundary is described by discontinuities of PCLS functions. The PCLS method is related to the phase‐field methods, and the topology optimization problem is defined as a minimization problem with piecewise constant constraints, without the need of solving the Hamilton–Jacobi equation. The result is not moving the boundaries during the iterative procedure. Thus, it offers some advantages in treating geometries, eliminating the reinitialization and naturally nucleating holes when needed. In the paper, the PCLS method is implemented with the additive operator splitting numerical scheme, and several numerical and procedural issues of the implementation are discussed. Examples of 2D structural topology optimization problem of minimum compliance design are presented, illustrating the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.

Influence of various design parameters on the quality of optimal shape design in topology optimization analysis

AbstractThe paper presents the discrete structural topology optimization problem. The analysis was made for the formulation of optimal design with a compliance minimization (as the objective functional) of statically loaded continuum structures with constraints forced on mass of the structure. Two different algorithms were compared (one taken from the literature and the second originally prepared) in order to study influence of design parameters formulation on the calculations results. Optimal topologies were presented and discussed for several 2D examples, with various optimization process parameters in order to get the best results in the shortest possible time. Other goal of presented research is to find out the possibilities of applying proposed algorithm in the designing process of bridge girders. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the applicability and solution of bilevel optimization models in transportation science: A study on the existence, stability and computation of optimal solutions to stochastic mathematical programs with equilibrium constraints

Bilevel optimization models, and more generally MPEC (mathematical program with equilibrium constraints) models, constitute important modelling tools in transportation science and network games, as they place the classic “what-if” analysis in a proper mathematical framework. The MPEC model is also becoming a standard for the computation of optimal design solutions, where “design” may include either or both of network infrastructure investments and various types of tolls. At the same time, it does normally not sufficiently well take into account possible uncertainties and/or perturbations in problem data (travel costs and demands), and thus may not a priori guarantee robust designs under varying conditions. We consider natural stochastic extensions to a class of MPEC traffic models which explicitly incorporate data uncertainty. In stochastic programming terminology, we consider “here-and-now” models where decisions on the design must be made before observing the uncertain parameter values and the responses of the network users, and the design is chosen to minimize the expectation of the upper-level objective function. Such a model could, for example, be used to derive a fixed link pricing scheme that provides the best revenue for a given network over a given time period, where the varying traffic conditions are described by distributions of parameters in the link travel time and OD demand functions. For a general such SMPEC network model we establish not only the existence of optimal solutions, but in particular their stability to perturbations in the probability distribution. We also provide convergence results for general algorithmic schemes based on the penalization of the equilibrium conditions or possible joint upper-level constraints, as well as for algorithms based on the discretization of the probability distribution, the latter enabling the utilization of standard MPEC algorithms. Especially the latter part utilizes relations between the traffic application of SMPEC and stochastic structural topology optimization problems.

Structural optimization under uncertain loads and nodal locations

This paper presents algorithms for solving structural topology optimization problems with uncertainty in the magnitude and location of the applied loads and with small uncertainty in the location of the structural nodes. The second type of uncertainty would typically arise from fabrication errors where the tolerances for the node locations are small in relation to the length scale of the structural elements. We first review the discrete form of the uncertain loads problem, which has been previously solved using a weighted average of multiple load patterns. With minor modifications, we extend this solution to include loads described by continuous joint probability density functions. We then proceed to the main contribution of this paper: structural optimization under uncertainty in the nodal locations. This optimization problem is computationally difficult because it involves variations of the inverse of the structural stiffness matrix. It is shown, however, that for small uncertainties the problem can be recast into a simpler but equivalent structural optimization problem with equivalent uncertain loads. By expressing these equivalent loads in terms of continuous random variables, we are able to make use of the extended form of the uncertain loads problem presented in the first part of this paper. The optimization algorithms are developed in the context of minimum compliance (maximum stiffness) design. Simple examples are presented. The results demonstrate that load and nodal uncertainties can have dramatic impact on optimal design. For structures containing thin substructures under axial loads, it is shown that these uncertainties (a) are of first-order significance, influencing the linear elastic response quantities, and (b) can affect designs by avoiding unrealistically optimistic and potentially unstable structures. The additional computational cost associated with the uncertainties scales linearly with the number of uncertainties and is insignificant compared to the cost associated with solving the deterministic structural optimization problem.

A level set method for reliability-based topology optimization of compliant mechanisms

Based on the level set model and the reliability theory, a numerical approach of reliability-based topology optimization for compliant mechanisms with multiple inputs and outputs is presented. A multi-objective topology optimal model of compliant mechanisms considering uncertainties of the loads, material properties, and member geometries is developed. The reliability analysis and topology optimization are integrated in the optimal iterative process. The reliabilities of the compliant mechanisms are evaluated by using the first order reliability method. Meanwhile, the problem of structural topology optimization is solved by the level set method which is flexible in handling complex topological changes and concise in describing the boundary shape of the mechanism. Numerical examples show the importance of considering the stochastic nature of the compliant mechanisms in the topology optimization process.