Evolutionary structural optimization for dynamic problems

This paper presents a simple method for structural optimization with frequency constraints. The structure is modelled by a fine mesh of finite elements. At the end of each eigenvalue analysis, part of the material is removed from the structure so that the frequencies of the resulting structure will be shifted towards a desired direction. A sensitivity number indicating the optimum locations for such material elimination is derived. This sensitivity number can be easily calculated for each element using the information of the eigenvalue solution. The significance of such an evolutionary structural optimization (ESO) method lies in its simplicity in achieving shape and topology optimization for both static and dynamic problems. In this paper, the ESO method is applied to a wide range of frequency optimization problems, which include maximizing or minimizing a chosen frequency of a structure, keeping a chosen frequency constant, maximizing the gap of arbitrarily given two frequencies, as well as considerations of multiple frequency constraints. The proposed ESO method is verified through several examples whose solutions may be obtained by other methods.     

Improving efficiency of evolutionary structural optimization by implementing fixed grid mesh

Evolutionary structural optimization (ESO) has been shown through much published research to be a simple and yet effective method for structural shape and topology optimization. However, attention has been drawn to shortcomings in the method related to the computational efficiency of the algorithm as well as the jagged edge representation of the Finite Element optimal solutions. In this paper a fixed grid (FG) mesh is implemented and an improved ESO methodology is introduced in order to address these shortcomings. The examples show a significant improvement in the solution time as well as eliminating the jagged edges and checkerboard patterns that may appear in current solution topologies. In addition, FG is applied to both stress based and stiffness optimization. This paper demonstrates a simple implementation of FG and the consequent improvement in the efficiency and practicality of the FG ESO formulation.     

Robust Input Covariance Constraint Control for Uncertain Polytopic Systems

In this paper, the robust input covariance constraint (ICC) control problem with polytopic uncertainty is solved using convex optimization with linear matrix inequality (LMI) approach. The ICC control problem is an optimal control problem that optimizes the output performance subjected to multiple constraints on the input covariance matrices. This control problem has significant practical implications when hard constraints need to be satisfied on control actuators. The contribution of this paper is the characterization of the control synthesis LMIs used to solve the robust ICC control problem for polytopic uncertain systems. Both continuous‐ and discrete‐time systems are considered. Parameter‐dependent and independent Lyapunov functions have been used for robust ICC controller synthesis. Numerical design examples are presented to illustrate the effectiveness of the proposed approach.     

The evolutionary structural optimization method: theoretical aspects

In 1993, Y.M. Xie and G.P. Steven introduced an approach called evolutionary structural optimization (ESO). ESO is based on the simple idea that the optimal structure (maximum stiffness, minimum weight) can be produced by gradually removing the ineffectively used material from the design domain. The design domain is constructed by the FE method, and furthermore, external loads and support conditions are applied to the element model. Considering the engineering aspects, ESO seems to have some attractive features: the ESO method is very simple to program via the FEA packages and requires a relatively small amount of FEA time. Additionally, the ESO topologies have been compared with analytical ones, e.g. Michell trusses, and so far the results are quite promising. On the other hand, ESO does not have a solid theoretical basis, and consequently, the ESO minimization problem is still unresolved. Since the good agreement between the results cannot be just a coincidence, in this paper, we will study whether the gradual removal of material can be explained mathematically and whether the theoretical basis of ESO can be outlined.First, a minimization problem solved by ESO is examined. Based on the results of earlier publications, it was assumed that the ESO method minimizes the compliance-volume product of a structure or a finite element model. It was noted that the sequential linear programming (SLP)-based approximate optimization method followed by the Simplex algorithm is equivalent to ESO if the strain energy rejection criterion is utilized. However, ESO should be applied so that the elements corresponding to the design domain are equally sized. If this requirement is not met, the rejection criterion, which also considers the varying sizes of the elements, should be used. Additionally, the element stiffness matrices and element volumes should be linearly dependent on the design variables. Also linearly elastic material is assumed. At each iteration the rejected elements should be removed completely. Most often only element removal is allowed in ESO. If the design variables are initially assigned values other than the maximum value, however, the elements should be allowed to reenter the design domain. This subject, obviously, needs further study. Typically, ESO is applied to problems having a planar design domain with in-plane forces only. In these cases, ESO produces truss-like, equally stressed and maximum-stiffness topologies. It is often recommended that, based on the topology optimization, a new finite element discretization should be employed. After that, the sizing optimization procedure can be performed. Since ESO seems to be producing truss-like topologies, ESO should be applied to structural problems having pin-jointed connections. For other types of structures ESO should be studied further.Finally, it can be concluded that ESO is not just an intuitive method, as it has a very distinct theoretical basis. It is also very simple to employ in engineering design problems. For this reason, ESO has potential to become a tool for design engineers.     

Evolutionary topology optimization of continuum structures with an additional displacement constraint

Evolutionary structural optimization (ESO) and its later version Bi-directional ESO (BESO) have been successfully applied to optimum material distribution problems for continuum structures. However, the existing ESO/BESO methods are limited to the topology optimization of an objective function such as mean compliance with a single constraint e.g. structural volume. The present work extends the BESO method to the stiffness optimization with a material volume constraint and a local displacement constraint. As a result, one will obtain a structure with the highest stiffness for a given volume while the displacement of a certain node does not exceed a prescribed limit. Several examples are presented to demonstrate the effectiveness of the proposed method.     

An algebraic approach to shape-from-image problems

This paper presents a new method for recovering three-dimensional shapes of polyhedral objects from their single-view images. The problem of recovery is formulated in a constrained optimization problem, in which the constraints reflect the assumption that the scene is composed of polyhedral objects, and the objective function to be minimized is a weighted sum of quadric errors of surface information such as shading and texture. For practical purpose it is decomposed into the two more tractable problems: a linear programming problem and an unconstrained optimization problem. In the present method the global constraints placed by the polyhedron assumption are represented in terms of linear algebra, whereas similar constraints have usually been represented in terms of a gradient space. Moreover, superstrictness of the constraints can be circumvented by a new concept ‘position-free incidence structure’. For this reason the present method has several advantages: it can recover the polyhedral shape even if image data are incorrect due to vertex-position errors, it can deal with perspective projection as well as orthographic projection, the number of variables in the optimization problem is very small (three or a little greater than three), and any kinds of surface information can be incorporated in a unifying manner.     

A further review of ESO type methods for topology optimization

Evolutionary Structural Optimization (ESO) and its later version bi-directional ESO (BESO) have gained widespread popularity among researchers in structural optimization and practitioners in engineering and architecture. However, there have also been many critical comments on various aspects of ESO/BESO. To address those criticisms, we have carried out extensive work to improve the original ESO/BESO algorithms in recent years. This paper summarizes latest developments in BESO for stiffness optimization problems and compares BESO with other well-established optimization methods. Through a series of numerical examples, this paper provides answers to those critical comments and shows the validity and effectiveness of the evolutionary structural optimization method.     

The ESO method revisited

This paper examines the evolutionary structural optimisation (ESO) method and its shortcomings. By proposing a problem statement for ESO followed by an accurate sensitivity analysis a framework is presented in which ESO is mathematically justifiable. It is shown that when using a sufficiently accurate sensitivity analysis, ESO method is not prone to the problem proposed by Zhou and Rozvany (Struct Multidiscip Optim 21(1):80–83, 2001). A complementary discussion on previous communications about ESO and strategies to overcome the Zhou-Rozvany problem is also presented. Finally it is shown that even the proposed rigorous ESO approach can result in highly inefficient local optima. It is discussed that the reason behind this shortcoming is ESO’s inherent unidirectional approach. It is thus concluded that the ESO method should only be used on a very limited class of optimisation problems where the problem’s constraints demand a unidirectional approach to final solutions. It is also discussed that the Bidirectional ESO (BESO) method is not prone to this shortcoming and it is suggested that the two methods be considered as completely separate optimisation techniques.     

Using genetic algorithms to improve the thermodynamic efficiency of gas turbines designed by traditional methods

A method for optimizing the thermodynamic efficiency of aeronautical gas turbines designed by classical methods is presented. This method is based in the transformation of the original constrained optimization problem into a new constrained free optimization problem which is solved by a genetic algorithm. Basically, a set of geometric, aerodynamic and acoustic noise constraints must be fulfilled during the optimization process. As a case study, the thermodynamic efficiency of an already optimized by traditional methods real aeronautical low pressure turbine design of 13 rows has been successfully improved, increasing the turbine efficiency by 0.047% and reducing the total number of airfoils by 1.61%. In addition, experimental evidence of a strong correlation between the total number of airfoils and the turbine efficiency has been observed. This result would allow us to use the total number of airfoils as a cheap substitute of the turbine efficiency for a coarse optimization at the first design steps.     

Constraint aggregation for large number of constraints in wing surrogate-based optimization

The method of aggregating a large number of constraints into one or few constraints has been successfully applied to wing structural design using gradient-based local optimization. However, numerical difficulties may occur in the case that the local curvatures of the aggregated constraint become extremely large and then ill-conditioned Hessian matrix may be yielded. This paper aims to test different methods of constraint aggregation within the framework of a gradient-free optimization, which makes use of cheap-to-evaluate surrogate models to find the global optimum. Three constraint aggregation approaches are investigated: the maximum constraint approach, the constant parameter Kreisselmeier-Steinhauser (KS) function, and the adaptive KS function. We also explore methods of aggregating constraints over the entire structure and within sub-domains. Examples of structural optimization and aero-structural optimization for a transport aircraft wing are employed and the results show that (1) the KS function with a larger constant parameter ρ can lead to better optimization results than the adaptive method, as the active constraints are approximated more accurately; (2) lumping the constraints within sub-domains instead of all together can improve the accuracy of the aggregated constraint and therefore helps find a better design. Finally, it is concluded from current test cases that the most efficient way of handling large-scale constraints for wing surrogate-based optimization is to aggregate constraints within sub-domains and with a relatively large constant parameter.

     

A distributed chromosome genetic algorithm for bin-packing

Genetic algorithms (GAs) are excellent approaches to solving complex problems in optimization with difficult constraints. The classic bin-packing optimization problem has been shown to be a NP-complete problem. The loading multiple parts into the build cylinder of a rapid prototyping machine is a type of a bin-packing optimization problem. There are GA applications that work with variations of the bin-packing problem, such as stock cutting, vehicle loading, air container loading, scheduling, and knapsack problems. These applications are mostly based on one-dimensional or two-dimensional considerations, using very specific assumptions. Ikonen et al. (A genetic algorithm for packing three-dimensional non-convex objects having cavities and holes. In: Proceedings of the Seventh International Conference on Genetic Algorithms. Michigan State University, July 19–23, 1997.) have developed a GA for rapid prototyping called GARP, which utilizes a three-dimensional chromosome structure for the bin-packing of a Sinterstation 2000s build cylinder. GARP allows the Sinterstation 2000 to be used more productively by designing a packing method for multiple parts. GARP was developed using a sequential GA, so execution time is influenced by the number of parts to be packed. Anticipating greater use of time compression technologies, GARP's execution time needs to be reduced. This paper will detail the development of a distributed chromosome GA to reduce the execution time of GARP. The implementation of this distributed GA will improve the efficiency of GARP, by using multiple CPUs to help solve the problem of bin-packing the build cylinder for the rapid prototyping machine.     

A review of optimization of cast parts using topology optimization

In recent years, there has been considerable progress in the optimization of cast parts with respect to strength, stiffness, and frequency. Here, topology optimization has been the most important tool in finding the optimal features of a cast part, such as optimal cross-section or number and arrangement of ribs. An optimization process with integrated topology optimization has been used very successfully at Adam Opel AG in recent years, and many components have been optimized. This two-paper review gives an overview of the application and experience in this area. This is the first part of a two-paper review of optimization of cast parts.Here, we want to focus on the application of the original topology optimization codes, which do not take manufacturing constraints for cast parts into account. Additionally, the role of shape optimization as a fine-tuning tool will be briefly analyzed and discussed.     

Evolutionary level set method for structural topology optimization

This paper proposes an evolutionary accelerated computational level set algorithm for structure topology optimization. It integrates the merits of evolutionary structure optimization (ESO) and level set method (LSM). Traditional LSM algorithm is largely dependent on the initial guess topology. The proposed method combines the merits of ESO techniques with those of LSM algorithm, while allowing new holes to be automatically generated in low strain energy within the nodal neighboring region during optimization. The validity and robustness of the new algorithm are supported by some widely used benchmark examples in topology optimization. Numerical computations show that optimization convergence is accelerated effectively.     

QoS decomposition for service composition using genetic algorithm

Recently, a lot of research has been dedicated to optimizing the QoS-aware service composition. This aims at selecting the optimal composed service from all possible service combinations regarding user's end-to-end quality requirements. Existing solutions often employ the global optimization approach, which does not show promising performance. Also, the complexity of such methods extensively depends on the number of available web-services, which continuously increase along with the growth of the Internet. Besides, the local optimization approaches have been rarely utilized, since they may violate the global constraints. In this paper, we propose a top-down structure, named quality constraints decomposition (QCD) here, to decompose the global constraints into the local constraints, using the genetic algorithm (GA). Then the best web service for each task is selected through a simple linear search. In contrast to existing methods, the QCD approach mainly depends on a limited set of tasks, which is considerably less complex, especially in the case of dynamically distributed service composition. Experimental results, based on a well-known data set of web services (QWSs), show the advantages of the QCD method in terms of computation time, considering the number of web services.     

3D and multiple load case bi-directional evolutionary structural optimization (BESO)

Evolutionary Structural Optimization (ESO), is a numerical method of structural optimization that is integrated with finite element analysis (FEA). Bi-directional ESO (BESO) is an extension to this method and can begin with minimal amount of material (only that necessary to support the load and support cases) in contrast to ESO which uses an initially oversized structure. Using BESO the structure is then allowed to grow into the optimum design or shape by both adding elements where the stresses are the highest and taking elements away where stresses are the lowest. In conducting this research, a methodology was developed (and integrated into the ESO program EVOLVE) which produced the optimal 3D finite element models of a structure in a more reliable way than the traditional ESO method. Additionally, the BESO method was successfully extended to multiple load cases for both 2D and 3D. Two different algorithms were used to find the best structure experiencing more than one load case and the results of each are included.     

Enhanced POD projection basis with application to shape optimization of car engine intake port

In this paper we present a rigorous method for the construction of enhanced Proper Orthogonal Decomposition (POD) projection bases for the development of efficient Reduced Order Models (ROM). The resulting ROMs are seen to exactly interpolate global quantities by design, such as the objective function(s) and nonlinear constraints involved in the optimization problem, thus narrowing the search space by limiting the number of constraints that need to be explicitly included in the statement of the optimization problem. We decompose the basis into two subsets of orthogonal vectors, one for the representation of constraints and the other one, in a complementary space, for the minimization of the projection errors. An explicit algorithm is presented for the case of linear objective functions. The proposed method is then implemented within a bi-level ROM and is illustrated with an application to the multi-objective shape optimization of a car engine intake port for two competing objectives: CO₂ emissions and engine power. We show that optimization using the proposed method produces Pareto dominant and realistic solutions for the flow fields within the combustion chamber, providing further insight into the flow properties.     

An adaptive approach to constraint aggregation using adjoint sensitivity analysis

Constraint aggregation is the key for efficient structural optimization when using a gradient-based optimizer and an adjoint method for sensitivity analysis. We explore different methods of constraint aggregation for numerical optimization. We analyze existing approaches, such as considering all constraints individually, taking the maximum of the constraints and using the Kreisselmeier–Steinhauser (KS) function. A new adaptive approach based on the KS function is proposed that updates the aggregation parameter by taking into account the constraint sensitivity. This adaptive approach is shown to significantly increase the accuracy of the results without additional computational cost especially when a large number of constraints are active at the optimum. The characteristics of each aggregation method and the performance of the proposed adaptive approach are shown by solving a wing structure weight minimization problem.     

A new energy landscape paving heuristic for satellite module layouts

This article describes a study of the satellite module layout problem (SMLP), which is a three-dimensional (3D) layout optimization problem with performance constraints that has proved to be non-deterministic polynomial-time hard (NP-hard). To deal with this problem, we convert it into an unconstrained optimization problem using a quasi-physical strategy and the penalty function method. The energy landscape paving (ELP) method is a class of Monte-Carlo-based global optimization algorithm that has been successfully applied to solve many optimization problems. ELP can search for low-energy layouts via a random walk in complex energy landscapes. However, when ELP falls into the narrow and deep valleys of an energy landscape, it is difficult to escape. By putting forward a new update mechanism of the histogram function in ELP, we obtain an improved ELP method which can overcome this drawback. By incorporating the gradient method with local search into the improved ELP method, a new global search optimization method, nELP, is proposed for SMLP. Two representative instances from the literature are tested. Computational results show that the proposed nELP algorithm is an effective method for solving SMLP with performance constraints.     

Hybrid differential evolution and Nelder�CMead algorithm with re-optimization

Nonlinear optimization algorithms could be divided into local exploitation methods such as Nelder–Mead (NM) algorithm and global exploration ones, such as differential evolution (DE). The former searches fast yet could be easily trapped by local optimum, whereas the latter possesses better convergence quality. This paper proposes hybrid differential evolution and NM algorithm with re-optimization, called as DE-NMR. At first a modified NM, called NMR is presented. It re-optimizes from the optimum point at the first time and thus being able to jump out of local optimum, exhibits better properties than NM. Then, NMR is combined with DE. To deal with equal constraints, adaptive penalty function method is adopted in DE-NMR, which relaxes equal constraints into unequal constrained functions with an adaptive relaxation parameter that varies with iteration. Benchmark optimization problems as well as engineering design problems are used to experiment the performance of DE-NMR, with the number of function evaluation times being employed as the main index of measuring convergence speed, and objective function values as the main index of optimum’s quality. Non-parametric tests are employed in comparing results with other global optimization algorithms. Results illustrate the fast convergence speed of DE-NMR.