Showing papers in "Structural and Multidisciplinary Optimization in 2006"

Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation

Abstract: This paper presents an adaptive weighted sum (AWS) method for multiobjective optimization problems. The method extends the previously developed biobjective AWS method to problems with more than two objective functions. In the first phase, the usual weighted sum method is performed to approximate the Pareto surface quickly, and a mesh of Pareto front patches is identified. Each Pareto front patch is then refined by imposing additional equality constraints that connect the pseudonadir point and the expected Pareto optimal solutions on a piecewise planar hypersurface in the \( {m} \)-dimensional objective space. It is demonstrated that the method produces a well-distributed Pareto front mesh for effective visualization, and that it finds solutions in nonconvex regions. Two numerical examples and a simple structural optimization problem are solved as case studies.

Sequential kriging optimization using multiple-fidelity evaluations

Abstract: When cost per evaluation on a system of interest is high, surrogate systems can provide cheaper but lower-fidelity information. In the proposed extension of the sequential kriging optimization method, surrogate systems are exploited to reduce the total evaluation cost. The method utilizes data on all systems to build a kriging metamodel that provides a global prediction of the objective function and a measure of prediction uncertainty. The location and fidelity level of the next evaluation are selected by maximizing an augmented expected improvement function, which is connected with the evaluation costs. The proposed method was applied to test functions from the literature and a metal-forming process design problem via finite element simulations. The method manifests sensible search patterns, robust performance, and appreciable reduction in total evaluation cost as compared to the original method.

Topology optimization of heat conduction problems using the finite volume method

Abstract: This note addresses the use of the finite volume method (FVM) for topology optimization of a heat conduction problem Issues pertaining to the proper choice of cost functions, sensitivity analysis, and example test problems are used to illustrate the effect of applying the FVM as an analysis tool for design optimization This involves an application of the FVM to problems with nonhomogeneous material distributions, and the arithmetic and harmonic averages have here been used to provide a unique value for the conductivity at element boundaries It is observed that when using the harmonic average, checkerboards do not form during the topology optimization process

An augmented Lagrangian relaxation for analytical target cascading using the alternating direction method of multipliers

Abstract: Analytical target cascading is a method for design optimization of hierarchical, multilevel systems. A quadratic penalty relaxation of the system consistency constraints is used to ensure subproblem feasibility. A typical nested solution strategy consists of inner and outer loops. In the inner loop, the coupled subproblems are solved iteratively with fixed penalty weights. After convergence of the inner loop, the outer loop updates the penalty weights. The article presents an augmented Lagrangian relaxation that reduces the computational cost associated with ill-conditioning of subproblems in the inner loop. The alternating direction method of multipliers is used to update penalty parameters after a single inner loop iteration, so that subproblems need to be solved only once. Experiments with four examples show that computational costs are decreased by orders of magnitude ranging between 10 and 1000.

Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition

Abstract: This paper describes a phase field method for the optimization of multimaterial structural topology with a generalized Cahn–Hilliard model. Similar to the well-known simple isotropic material with penalization method, the mass concentration of each material phase is considered as design variable. However, a variational approach is taken with the Cahn–Hilliard theory to define a thermodynamic model, taking into account of the bulk energy and interface energy of the phases and the elastic strain energy of the structure. As a result, the structural optimization problem is transformed into a phase transition problem defined by a set of nonlinear parabolic partial differential equations. The generalized Cahn–Hilliard model regularizes the original ill-posed topology optimization problem and provides flexibility of topology changes with interface coalescence and break-up due to phase separation and coarsening. We employ a powerful multigrid algorithm and extend it to include four material phases for numerical solution of the Cahn–Hilliard equations. We demonstrate our approach through several 2-D and 3-D examples to minimize mean compliance of the multimaterial structures.

A review of optimization of structures subjected to transient loads

Abstract: Various aspects of structural optimization techniques under transient loads are extensively reviewed. The main themes of the paper are treatment of time-dependent constraints, calculation of design sensitivity, and approximation. Each subject is reviewed with corresponding papers that have been published since the 1970s. The treatment of time-dependent constraints in both the direct method and the transformation method is discussed. Two ways of calculating design sensitivity of a structure under transient loads are discussed—direct differentiation method and adjoint variable method. The approximation concept mainly focuses on the response surface method in crashworthiness and local approximation with the intermediate variables. Especially, a method using the equivalent static load is discussed as an approximation method. It takes advantage of the well-established static response optimization. The structural optimization in flexible multibody dynamic systems is reviewed in the viewpoint of the above three themes.

A direct decoupling approach for efficient reliability-based design optimization

Abstract: This paper develops an efficient methodology to perform reliability-based design optimization (RBDO) by decoupling the optimization and reliability analysis iterations that are nested in traditional formulations. This is achieved by approximating the reliability constraints based on the reliability analysis results. The proposed approach does not use inverse first-order reliability analysis as other existing decoupled approaches, but uses direct reliability analysis. This strategy allows a modular approach and the use of more accurate methods, including Monte-Carlo-simulation (MCS)-based methods for highly nonlinear reliability constraints where first-order reliability approximation may not be accurate. The use of simulation-based methods also enables system-level reliability estimates to be included in the RBDO formulation. The efficiency of the proposed RBDO approach is further improved by identifying the potentially active reliability constraints at the beginning of each reliability analysis. A vehicle side impact problem is used to examine the proposed method, and the results show the usefulness of the proposed method.

Update strategies for kriging models used in variable fidelity optimization

Abstract: Many optimization methods for simulation-based design rely on the sequential use of metamodels to reduce the associated computational burden. In particular, kriging models are frequently used in variable fidelity optimization. Nevertheless, such methods may become computationally inefficient when solving problems with large numbers of design variables and/or sampled data points due to the expensive process of optimizing the kriging model parameters in each iteration. One solution to this problem would be to replace the kriging models with traditional Taylor series response surface models. Kriging models, however, were shown to provide good approximations of computer simulations that incorporate larger amounts of data, resulting in better global accuracy. In this paper, a metamodel update management scheme (MUMS) is proposed to reduce the cost of using kriging models sequentially by updating the kriging model parameters only when they produce a poor approximation. The scheme uses the trust region ratio (TR-MUMS), which is a ratio that compares the approximation to the true model. Two demonstration problems are used to evaluate the proposed method: an internal combustion engine sizing problem and a control-augmented structural design problem. The results indicate that the TR-MUMS approach is very effective; on the demonstration problems, it reduced the number of likelihood evaluations by three orders of magnitude compared to using a global optimizer to find the kriging parameters in every iteration. It was also found that in trust region-based method, the kriging model parameters need not be updated using a global optimizer—local methods perform just as well in terms of providing a good approximation without affecting the overall convergence rate, which, in turn, results in a faster execution time.

Multiobjective evolutionary optimization of periodic layered materials for desired wave dispersion characteristics

Abstract: An important dispersion-related characteristic of wave propagation through periodic materials is the existence of frequency bands. A medium effectively attenuates all inci- dent waves within stopbands and allows propagation within passbands. The widths and locations of these bands in the frequency domain depend on the layout of contrasting mate- rials and the ratio of their properties. Using a multiobjective genetic algorithm, the topologies of one-dimensional peri- odic unit cells are designed for target frequency band struc- tures characterizing longitudinal wave motion. The decision variables are the number of layers in the unit cell and the thickness of each layer. Binary and mixed formulations are developed for the treatment of the optimization problems. Designs are generated for the following novel objectives: (1) maximum attenuation of time harmonic waves, (2) maximum isolation of general broadband pulses, and (3) filtering sig- nals at predetermined frequency windows. The saturation of performance with the number of unit-cell layers is shown for the first two cases. In the filtering application, the trade-off between the simultaneous realization of passband and stop- band targets is analyzed. It is shown that it is more difficult to design for passbands than it is to design for stopbands. The design approach presented has potential use in the de- velopment of vibration and shock isolation structures, sound isolation pads/partitions, and multiple band frequency filters, among other applications.

Maximizing band gaps in plate structures

Abstract: Band gaps, i.e., frequency ranges in which waves cannot propagate, can be found in elastic structures for which there is a certain periodic modulation of the material properties or structure. In this paper, we maximize the band gap size for bending waves in a Mindlin plate. We analyze an infinite periodic plate using Bloch theory, which conveniently reduces the maximization problem to that of a single base cell. Secondly, we construct a finite periodic plate using a number of the optimized base cells in a postprocessed version. The dynamic properties of the finite plate are investigated theoretically and experimentally and the issue of finite size effects is addressed.

Performance-based multiobjective optimum design of steel structures considering life-cycle cost

Abstract: A new methodology for the performance-based optimum design of steel structures subjected to seismic loading considering inelastic behavior is proposed. The importance of considering life-cycle cost as an additional objective to the initial structural cost objective function in the context of multiobjective optimization is also investigated. Life-cycle cost is considered to take into account during the design phase the impact of future earthquakes. For the solution of the multiobjective optimization problem, Evolutionary Algorithms and in particular an algorithm based on Evolution Strategies, specifically tailored to meet the characteristics of the problem at hand, are implemented. The constraints of the optimization problem are based on the provisions of European design codes, while additional constraints are imposed by means of pushover analysis to control the load and deformation capacity of the structure.

Fully stressed design of passive controllers in framed structures for seismic loadings

Abstract: This paper addresses the optimal design problem of added damping in framed structures. Interstory performance indices for linear and nonlinear structures are chosen and restricted to allowable values under the excitation of an ensemble of realistic ground motion records. Optimality criteria are formulated based on fully stressed characteristics of the optimal solution, and a simple analysis/redesign procedure is proposed for attaining optimal designs. Results of four examples presented compare well to those obtained using formal gradient-based optimization.

Optimization of natural frequencies of bidirectional functionally graded beams

Abstract: We propose a methodology to optimize the natural frequencies of functionally graded structures by tailoring their material distribution. The element-free Galerkin method is used to analyze the two-dimensional steady-state free and forced vibration of functionally graded beams. To optimize the material composition, the spatial distribution of volume fractions of the material constituents is defined using piecewise bicubic interpolation of volume fraction values that are specified at a finite number of grid points. Subsequently, we use a real-coded genetic algorithm to optimize the volume fraction distribution for three model problems. In the first problem, we seek material distributions that maximize each of the first three natural frequencies of a functionally graded beam. The goal of the second model problem is to minimize the mass of a functionally graded beam while constraining its natural frequencies to lie outside certain prescribed frequency bands. The last problem aims to minimize the mass of a functionally graded beam by simultaneously optimizing its thickness and material distribution such that the fundamental frequency is greater than a prescribed value.

Using augmented Lagrangian particle swarm optimization for constrained problems in engineering">Using augmented Lagrangian particle swarm optimization for constrained problems in engineering

Abstract: The comparatively new stochastic method of particle swarm optimization (PSO) has been applied to engineering problems especially of nonlinear, non-differentiable, or non-convex type. Its robustness and its simple applicability without the need for cumbersome derivative calculations make PSO an attractive optimization method. However, engineering optimization tasks often consist of problem immanent equality and inequality constraints which are usually included by inadequate penalty functions when using stochastic algorithms. The simple structure of basic particle swarm optimization characterized by only a few lines of computer code allows an efficient implementation of a more sophisticated treatment of such constraints. In this paper, we present an approach which utilizes the simple structure of the basic PSO technique and combines it with an extended non-stationary penalty function approach, called augmented Lagrange multiplier method, for constraint handling where ill conditioning is a far less harmful problem and the correct solution can be obtained even for finite penalty factors. We describe the basic PSO algorithm and the resulting method for constrained problems as well as the results from benchmark tests. An example of a stiffness optimization of an industrial hexapod robot with parallel kinematics concludes this paper and shows the applicability of the proposed augmented Lagrange particle swarm optimization to engineering problems.

Optimization with non-homogeneous failure criteria like Tsai–Wu for composite laminates

Abstract: In designing composite laminates, minimization of a suitable failure criterion is sometimes selected as the objective function. However, for non-homogeneous criteria, e.g., the Tsai–Wu criterion, this objective function will not maximize the failure load, when it is carried at a load which is different from the failure load. We suggest that the use of a safety factor for the objective function is more appropriate for maximizing for the faiure load. In fact we show losses of more than 40% in the load carrying capacity even when the load carrying capacity of the optimal laminate is 75% of the applied load.

Structural optimization with $\tt{FreeFem++}$

Abstract: The aim of this paper is to show that relatively small, simple, and efficient shape optimization routines can be written using the free finite element software $\tt{FreeFem++}$ . This is illustrated by the implementation of two classical methods: the boundary variation method and the homogenization one. Even though these routines are simple enough so that their implementation can be assigned (partially or totally) as homework to graduate students, they yield results accurate enough to be useful tools for engineers or researchers.

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

Abstract: 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–Bolkow–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/ .

Reliability-based structural optimization of frame structures for multiple failure criteria using topology optimization techniques

Abstract: Topology optimization methods using discrete elements such as frame elements can provide useful insights into the underlying mechanics principles of products; however, the majority of such optimizations are performed under deterministic conditions. To avoid performance reductions due to later-stage environmental changes, variations of several design parameters are considered during the topology optimization. This paper concerns a reliability-based topology optimization method for frame structures that considers uncertainties in applied loads and nonstructural mass at the early conceptual design stage. The effects that multiple criteria, namely, stiffness and eigenfrequency, have upon system reliability are evaluated by regarding them as a series system, where mode reliabilities can be evaluated using first-order reliability methods. Through numerical calculations, reliability-based topology designs of typical two- or three-dimensional frames are obtained. The importance of considering uncertainties is then demonstrated by comparing the results obtained by the proposed method with deterministic optimal designs.

Penalty function approach for the mixed discrete nonlinear problems by particle swarm optimization

Abstract: In this paper, the basic characteristics of particle swarm optimization (PSO) for the global search are discussed at first, and then the PSO for the mixed discrete nonlinear problems (MDNLP) is suggested. The penalty function approach to handle the discrete design variables is employed, in which the discrete design variables are handled as the continuous ones by penalizing at the intervals. As a result, a useful method to determine the penalty parameter of penalty term for the discrete design variables is proposed. Through typical mathematical and structural optimization problems, the validity of the proposed approach for the MDNLP is examined.

Multidisciplinary and multiple operating points shape optimization of three-dimensional compressor blades

Abstract: The recent progress in simulation technologies in several fields such as computational fluid dynamics, structures, thermal analysis, and unsteady flow combined with the emergence of improved optimization algorithms makes it now possible to develop and use automatic optimization software and methodologies to perform complex multidisciplinary shape optimization process. In the present applications, the MAX optimization software developed at CENAERO is used to perform the optimization. This software allows performing derivative free optimization with very few calls to the computer intensive simulation software. The method employed in this paper combines the use of a genetic algorithm (with real coding of the variables) to an approximate (or meta) model to accelerate significantly the optimization process. The performance of this optimization methodology is illustrated on the optimization of three-dimensional turbomachinery blades for multiple operating points and multidisciplinary objectives and constraints. The NASA rotor 67 geometry is used to demonstrate the capabilities of the method. The aim is to find the optimal shape for three different operating conditions: one at a near peak efficiency point, one at choked mass flow, and one near the stall flow. The three points are analyzed at the same blade rotational speed but with different mass flows. A finite element structural mechanics software is used to compute the static and dynamic mechanical responses of the blade. A Navier–Stokes solver is used to calculate the aerodynamic performance. High performance computers (HPC) are used in this application. Cenaero’s HPC infrastructure contains a Linux cluster with 170 3.06 GHz Xeon processors. The optimization algorithm is parallelized using MPI.

Topology optimization for minimum mass design considering local failure constraints and contact boundary conditions

Abstract: The purpose of this work is to present a possible approach to the topology mass minimization of a body submitted to local material failure constraints, contact boundary conditions, and multiple load cases. The formulation combines the well-known SIMP approach (Solid Isotropic Microstructure with intermediate mass Penalization) and the Augmented Lagrangian technique to deal with stress-based constraints. At every design step and load case, a contact solver is called to obtain the equilibrium deformed configuration. Assuming differentiability, the sensitivity analysis is performed analytically at the cost of a Newton iteration. Finally, some numerical examples are presented to explore the differences and similarities found in the final designs for this case and for the case of minimization of internal energy, also with contact boundary conditions.

Topology optimization of 2D continua for minimum compliance using parallel computing

Abstract: Topology optimization is often used in the conceptual design stage as a preprocessing tool to obtain overall material distribution in the solution domain. The resulting topology is then used as an initial guess for shape optimization. It is always desirable to use fine computational grids to obtain high-resolution layouts that minimize the need for shape optimization and postprocessing (Bendsoe and Sigmund, Topology optimization theory, methods and applications. Springer, Berlin Heidelberg New York 2003), but this approach results in high computation cost and is prohibitive for large structures. In the present work, parallel computing in combination with domain decomposition is proposed to reduce the computation time of such problems. The power law approach is used as the material distribution method, and an optimality criteria-based optimizer is used for locating the optimum solution [Sigmund (2001)21:120–127; Rozvany and Olhoff, Topology optimization of structures and composites continua. Kluwer, Norwell 2000]. The equilibrium equations are solved using a preconditioned conjugate gradient algorithm. These calculations have been done using a master–slave programming paradigm on a coarse-grain, multiple instruction multiple data, shared-memory architecture. In this study, by avoiding the assembly of the global stiffness matrix, the memory requirement and computation time has been reduced. The results of the current study show that the parallel computing technique is a valuable tool for solving computationally intensive topology optimization problems.

Maximization of structural natural frequency with optimal support layout

Abstract: The optimal layout of supports is one of the key factors that dominates static and dynamic performances of the structure. In this work, supports are considered as elastic springs. The purpose is to carry out layout optimization of supports by means of topology optimization method. The technique of pseudo-density variables that transforms a discrete-variable problem into a continuous one is used in order that the problem is easily formulated and solved numerically. In this formulation, a power law of the so-called solid isotropic material with penalty model is employed to approximate the relation between the element stiffness matrix and density variable. Such a relation makes it easy to establish the computing scheme and sensitivity analysis of natural frequency. Support layout design that corresponds to optimization of boundary conditions is studied to maximize the natural frequency of structures. Numerical results show that reasonable distributions of supports can be generated effectively.

An efficient strategy for reliability-based multidisciplinary design optimization using BLISS

Abstract: This paper presents an efficient reliability-based multidisciplinary design optimization (RBMDO) strategy. The conventional RBMDO has tri-level loops: the first level is an optimization in the deterministic space, the second one is a reliability analysis in the probabilistic space, and the third one is the multidisciplinary analysis. Since it is computationally inefficient when high-fidelity simulation methods are involved, an efficient strategy is proposed. The strategy [named probabilistic bi-level integrated system synthesis (ProBLISS)] utilizes a single-level reliability-based design optimization (RBDO) approach, in which the reliability analysis and optimization are conducted in a sequential manner by approximating limit state functions. The single-level RBDO is associated with the BLISS formulation to solve RBMDO problems. Since both the single-level RBDO and BLISS are mainly driven by approximate models, the accuracy of models can be a critical issue for convergence. The convergence of the strategy is guaranteed by employing the trust region–sequential quadratic programming framework, which validates approximation models in the trust region radius. Two multidisciplinary problems are tested to verify the strategy. ProBLISS significantly reduces the computational cost and shows stable convergence while maintaining accuracy.

Sensitivity analysis of structural response uncertainty propagation using evidence theory

Abstract: Sensitivity analysis for the quantified uncertainty in evidence theory is developed. In reliability quantification, classical probabilistic analysis has been a popular approach in many engineering disciplines. However, when we cannot obtain sufficient data to construct probability distributions in a large-complex system, the classical probability methodology may not be appropriate to quantify the uncertainty. Evidence theory, also called Dempster–Shafer Theory, has the potential to quantify aleatory (random) and epistemic (subjective) uncertainties because it can directly handle insufficient data and incomplete knowledge situations. In this paper, interval information is assumed for the best representation of imprecise information, and the sensitivity analysis of plausibility in evidence theory is analytically derived with respect to expert opinions and structural parameters. The results from the sensitivity analysis are expected to be very useful in finding the major contributors for quantified uncertainty and also in redesigning the structural system for risk minimization.

Evolutionary truss topology optimization using a graph-based parameterization concept

Abstract: 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, ie, mutation and crossover, on them For this purpose, new genetic graph operators are introduced, which are combined with graph algorithms, eg, 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

Topology optimization in crashworthiness design

Abstract: Topology optimization has developed rapidly, primarily with application on linear elastic structures subjected to static loadcases. In its basic form, an approximated optimization problem is formulated using analytical or semi-analytical methods to perform the sensitivity analysis. When an explicit finite element method is used to solve contact–impact problems, the sensitivities cannot easily be found. Hence, the engineer is forced to use numerical derivatives or other approaches. Since each finite element simulation of an impact problem may take days of computing time, the sensitivity-based methods are not a useful approach. Therefore, two alternative formulations for topology optimization are investigated in this work. The fundamental approach is to remove elements or, alternatively, change the element thicknesses based on the internal energy density distribution in the model. There is no automatic shift between the two methods within the existing algorithm. Within this formulation, it is possible to treat nonlinear effects, e.g., contact–impact and plasticity. Since no sensitivities are used, the updated design might be a step in the wrong direction for some finite elements. The load paths within the model will change if elements are removed or the element thicknesses are altered. Therefore, care should be taken with this procedure so that small steps are used, i.e., the change of the model should not be too large between two successive iterations and, therefore, the design parameters should not be altered too much. It is shown in this paper that the proposed method for topology optimization of a nonlinear problem gives similar result as a standard topology optimization procedures for the linear elastic case. Furthermore, the proposed procedures allow for topology optimization of nonlinear problems. The major restriction of the method is that responses in the optimization formulation must be coupled to the thickness updating procedure, e.g., constraint on a nodal displacement, acceleration level that is allowed.

A hierarchical genetic algorithm with age structure for multimodal optimal design of hybrid composites

Abstract: A genetic algorithm aiming the optimal design of composite structures under non-linear behaviour is presented. The approach addresses the optimal material/stacking sequence in laminate construction and material distribution topology in composite structures as a multimodal optimization problem. The proposed evolutionary process is based on a sequential hierarchical relation between subpopulations evolving in separated isolation stages followed by migration. Improvements based on the species conservation paradigm are performed to avoid genetic tendencies due to elitist strategies used in the hierarchical subpopulations. The concept of species is associated with material distribution topology in composite structures, and an enlarged master population with age structure is considered concurrently with the hierarchical topology. Rules based on species concept are imposed on either isolation or migration stages to overcome the predominance of a species and to guarantee the diversity. A mutation process controlled by the stress field is implemented, improving the local genetic search. The proposed model allows multiple solutions for the optimal design problem.

Semi-Lagrange method for level-set-based structural topology and shape optimization

Abstract: In this paper, we introduce a semi-Lagrange scheme to solve the level-set equation in structural topology optimization. The level-set formulation of the problem expresses the optimization process as a solution to a Hamilton–Jacobi partial differential equation. It allows for the use of shape sensitivity to derive a speed function for a descent solution. However, numerical stability condition in the explicit upwind scheme for discrete level-set equation severely restricts the time step, requiring a large number of time steps for a numerical solution. To improve the numerical efficiency, we propose to employ a semi-Lagrange scheme to solve level-set equation. Therefore, a much larger time step can be obtained and a much smaller number of time steps are required. Numerical experiments comparing the semi-Lagrange method with the classical explicit upwind scheme are presented for the problem of mean compliance optimization in two dimensions.

Structural optimization for post-buckling behavior using particle swarms

Abstract: The aim of this paper is to develop a new algorithm based on the particle swarm optimization (PSO) concept and then to apply it in the solution of some new structural optimization problems for post-buckling behavior. Proposed modifications of the algorithm regard both the PSO kernel and the constraints handling. The “controlled reflection” technique is proposed for dealing with inequality constraints. The values of the objective are calculated for some control points chosen along a move vector. The position for which the objective is the smallest one and the constraints are not violated is selected. For the case of equality constraints, the “particle trap” strategy is proposed. First, equalities are transformed into inequalities forming constraint “zone of influence.” If a particle from a swarm drops into this “zone” it remains trapped there and can move further only inside this subspace. Simultaneously, a penalty term is added to the objective function to force particles to be “captured” and constraints to become active at the optimum. The new PSO algorithm has been successfully applied to problems of structural optimization against instability. The standard maximization of the critical load is performed both for single and double buckling loads. The modified optimization for post-buckling behavior is also performed. A new problem of reconstruction of a predicted post-buckling path is formulated. The sum of squared distances between the control points of a given equilibrium path and the reconstructed one is minimized. Another new problem regards the modification of the slope of nonlinear equilibrium curve. This is obtained by adding a set of post-buckling constraints imposed on derivative values calculated for selected control points at the equilibrium curve.