Algebraic multilevel iteration method for lowest order Raviart–Thomas space and applications

An optimal order algebraic multilevel iterative method for solving system of linear algebraic equations arising from the finite element discretization of certain boundary value problems, that have their weak formulation in the space H(div), is presented. The algorithm is developed for the discrete problem obtained by using the lowest‐order Raviart–Thomas space. The method is theoretically analyzed and supporting numerical examples are presented. Furthermore, as a particular application, the algorithm is used for the solution of the discrete minimization problem which arises in the functional‐type a posteriori error estimates for the discontinuous Galerkin approximation of elliptic problems. Copyright © 2011 John Wiley & Sons, Ltd.     

An optimization technique using the characteristics of genetic algorithm

Optimization problems could happen often in discrete or discontinuous search space. Therefore, the traditional gradient‐based methods are not able to apply to this kind of problems. The discrete design variables are considered reasonably and the heuristic techniques are generally adopted to solve this problem, and the genetic algorithm based on stochastic search technique is one of these. The genetic algorithm method with discrete variables can be applied to structural optimization problems, such as composite laminated structures or trusses. However, the discrete optimization adopted in genetic algorithm gives rise to a troublesome task that is a mapping between each strings and discrete variables. And also, its solution quality could be restricted in some cases. In this study, a technique using the genetic algorithm characteristics is developed to utilize continuous design variables instead of discrete design variables in discontinuous solution spaces. Additionally, the proposed algorithm, which is manipulating a fitness function artificially, is applied to example problems and its results are compared with the general discrete genetic algorithm. The example problems are to optimize support positions of an unstable structure with discontinuous solution spaces.     

Multi-objective optimization of engineering systems using game theory and particle swarm optimization

This study proposes particle swarm optimization (PSO) based algorithms to solve multi-objective engineering optimization problems involving continuous, discrete and/or mixed design variables. The original PSO algorithm is modified to include dynamic maximum velocity function and bounce method to enhance the computational efficiency and solution accuracy. The algorithm uses a closest discrete approach (CDA) to solve optimization problems with discrete design variables. A modified game theory (MGT) approach, coupled with the modified PSO, is used to solve multi-objective optimization problems. A dynamic penalty function is used to handle constraints in the optimization problem. The methodologies proposed are illustrated by several engineering applications and the results obtained are compared with those reported in the literature.     

Using a derivative-free optimization method for multiple solutions of inverse transport problems

Identifying unknown components of an object that emits radiation is an important problem for national and global security. Radiation signatures measured from an object of interest can be used to infer object parameter values that are not known. This problem is called an inverse transport problem. An inverse transport problem may have multiple solutions and the most widely used approach for its solution is an iterative optimization method. This paper proposes a stochastic derivative-free global optimization algorithm to find multiple solutions of inverse transport problems. The algorithm is an extension of a multilevel single linkage (MLSL) method where a mesh adaptive direct search (MADS) algorithm is incorporated into the local phase. Numerical test cases using uncollided fluxes of discrete gamma-ray lines are presented to show the performance of this new algorithm.     

Implementation of Discrete Capability into the Enhanced Multipoint Approximation Method for Solving Mixed Integer-Continuous Optimization Problems

Multipoint approximation method (MAM) focuses on the development of metamodels for the objective and constraint functions in solving a mid-range optimization problem within a trust region. To develop an optimization technique applicable to mixed integer-continuous design optimization problems in which the objective and constraint functions are computationally expensive and could be impossible to evaluate at some combinations of design variables, a simple and efficient algorithm, coordinate search, is implemented in the MAM. This discrete optimization capability is examined by the well established benchmark problem and its effectiveness is also evaluated as the discreteness interval for discrete design variables is increased from 0.2 to 1. Furthermore, an application to the optimization of a lattice composite fuselage structure where one of design variables (number of helical ribs) is integer is also presented to demonstrate the efficiency of this capability.     

缓冲区容量约束下发动机混流装配线排序研究

为解决缓冲区容量约束下发动机混流装配排序问题,以关键部件消耗均匀化和最大完工时间最小化为目标,建立了优化数学模型,设计了一种多目标遗传算法,采用了混合交叉算子和启发式变异方法,并设计了基于帕累托分级和共享函数的适应度函数,将多目标遗传算法和多目标模拟退火算法的优化结果进行了比较。研究结果表明,多目标遗传算法在满意度和计算效率方面均优于多目标模拟退火算法,是一种有效的混流装配线排序问题求解算法。     

An alternating optimization approach for mixed discrete non-linear programming

This article contributes to the development of the field of alternating optimization (AO) and general mixed discrete non-linear programming (MDNLP) by introducing a new decomposition algorithm (AO-MDNLP) based on the augmented Lagrangian multipliers method. In the proposed algorithm, an iterative solution strategy is proposed by transforming the constrained MDNLP problem into two unconstrained components or units; one solving for the discrete variables, and another for the continuous ones. Each unit focuses on minimizing a different set of variables while the other type is frozen. During optimizing each unit, the penalty parameters and multipliers are consecutively updated until the solution moves towards the feasible region. The two units take turns in evolving independently for a small number of cycles. The validity, robustness and effectiveness of the proposed algorithm are exemplified through some well known benchmark mixed discrete optimization problems.     

A new harmony search algorithm for solving mixed–discrete engineering optimization problems

In general design optimization problems, it is usually assumed that the design variables are continuous. However, many practical problems in engineering design require considering the design variables as integer or discrete values. The presence of discrete and integer variables along with continuous variables adds to the complexity of the optimization problem. Very few of the existing methods can yield a globally optimal solution when the objective functions are non-convex and non-differentiable. This article presents a mixed–discrete harmony search approach for solving these nonlinear optimization problems which contain integer, discrete and continuous variables. Some engineering design examples are also presented to demonstrate the effectiveness of the proposed method.     

Truss topology,shape and sizing optimization by fully stressed design based on hybrid grey wolf optimization and adaptive differential evolution

A hybrid adaptive optimization algorithm based on integrating grey wolf optimization into adaptive differential evolution with fully stressed design (FSD) local search is presented in this article. Hybrid reproduction and control parameter adaptation strategies are employed to increase the performance of the algorithm. The proposed algorithm, called fully stressed design–grey wolf–adaptive differential evolution (FSD-GWADE), is demonstrated to tackle a variety of truss optimization problems. The problems have mixed continuous/discrete design variables that are assigned as simultaneous topology, shape and sizing design variables. FSD-GWADE provides competitive results and gives superior results at a higher success rate than the previous FSD-based algorithm.     

Convex multilevel decomposition algorithms for non-monotone problems

A convex, multilevel decomposition algorithm is proposed in this paper for the solution of static analysis problems involving non-monotone, possibly multivalued laws. The theory is developed here for a model structure with non-monotone interface or boundary conditions. First the non-monotone laws are written in the form of a difference of two monotone functions. Under this decomposition, the non-linear elastostatic analysis problem is equivalent to a system of convex variational inequalities and to non-convex min-min problems for appropriately defined Lagrangian functions. The solution(s) of each one of the aforementioned problems describe the position(s) of static equilibrium of the considered structure. In this paper a multilevel optimization scheme, due to Auchmuty,¹ is used for the numerical solution of the problem. The most interesting feature of this method, from the computational mechanics' standpoint, is the fact that each one of the subproblems involved in the multilevel algorithm is a convex optimization problem, or, in terms of mechanics, an appropriately modified monotone ‘unilateral’ problem. Thus, existing algorithms and software can be used for the numerical solution with minor modifications. Numerical results concerning the calculation of elastic and rigid stamp problems and of material inclusion problems with delamination and non-monotone stick-slip frictional effects illustrate the theory.     

Level set topology optimization of fluids in Stokes flow

We propose the level set method of topology optimization as a viable, robust and efficient alternative to density‐based approaches in the setting of fluid flow. The proposed algorithm maintains the discrete nature of the optimization problem throughout the optimization process, leading to significant advantages over density‐based topology optimization algorithms. Specifically, the no‐slip boundary condition is implemented directly—this is accurate, removes the need for interpolation schemes and continuation methods, and gives significant computational savings by only requiring flow to be modeled in fluid regions. Topological sensitivity information is utilized to give a robust algorithm in two dimensions and familiar two‐dimensional power dissipation minimization problems are solved successfully. Computational efficiency of the algorithm is also clearly demonstrated on large‐scale three‐dimensional problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Multiobjective design of actively controlled structures using a hybrid optimization method

A multiobjective approach to the combined structure and control optimization problem for flexible space structures is presented. The proposed formulation addresses robustness considerations for controller design, as well as a simultaneous determination of optimum actuator locations. The structural weight, controlled system energy, stability robustness index and damping augmentation provided by the active controller are considered as objective functions of the multiobjective problem which is solved using a cooperative game-theoretic approach. The actuator locations and the cross-sectional areas of structural members are treated as design variables. Since the actuator locations are spatially discrete, whereas the cross-sectional areas are continuous, the optimization problem has mixed discrete-continuous design variables. A solution approach to this problem based on a hybrid optimization scheme is presented. The hybrid optimizer is a synergetic blend of artificial genetic search and gradient-based search techniques. The computational procedure is demonstrated through the design of an ACOSS-FOUR space structure. The optimum solutions obtained using the hybrid optimizer are shown to outperform the optimum results obtained using gradient-based search techniques.     

A hierarchical method for discrete structural topology design problems with local stress and displacement constraints

In this paper, we present a hierarchical optimization method for finding feasible true 0–1 solutions to finite‐element‐based topology design problems. The topology design problems are initially modelled as non‐convex mixed 0–1 programs. The hierarchical optimization method is applied to the problem of minimizing the weight of a structure subject to displacement and local design‐dependent stress constraints. The method iteratively treats a sequence of problems of increasing size of the same type as the original problem. The problems are defined on a design mesh which is initially coarse and then successively refined as needed. At each level of design mesh refinement, a neighbourhood optimization method is used to treat the problem considered. The non‐convex topology design problems are equivalently reformulated as convex all‐quadratic mixed 0–1 programs. This reformulation enables the use of methods from global optimization, which have only recently become available, for solving the problems in the sequence. Numerical examples of topology design problems of continuum structures with local stress and displacement constraints are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

An engineering method for complex structural optimization involving both size and topology design variables

This work presents an engineering method for optimizing structures made of bars, beams, plates, or a combination of those components. Corresponding problems involve both continuous (size) and discrete (topology) variables. Using a branched multipoint approximate function, which involves such mixed variables, a series of sequential approximate problems are constructed to make the primal problem explicit. To solve the approximate problems, genetic algorithm (GA) is utilized to optimize discrete variables, and when calculating individual fitness values in GA, a second-level approximate problem only involving retained continuous variables is built to optimize continuous variables. The solution to the second-level approximate problem can be easily obtained with dual methods. Structural analyses are only needed before improving the branched approximate functions in the iteration cycles. The method aims at optimal design of discrete structures consisting of bars, beams, plates, or other components. Numerical examples are given to illustrate its effectiveness, including frame topology optimization, layout optimization of stiffeners modeled with beams or shells, concurrent layout optimization of beam and shell components, and an application in a microsatellite structure. Optimization results show that the number of structural analyses is dramatically decreased when compared with pure GA while even comparable to pure sizing optimization.     

基于SA算法的行星齿轮减速器离散变量优化设计

考虑制造工艺要求,将所有设计变量均视为离散变量,包括一般离散变量和伪离散变量,并就这两种情况下状态产生函数的设计原理进行深入研究,解决了将模拟退火算法用于离散变量函数优化的关键技术问题,介绍了一种基于模拟退火算法的离散变量函数优化的新方法。行星齿轮传动中各齿轮的齿数受传动比条件、同轴条件和装配条件的限制而不能任意取值,齿轮的模数也要受国家标准的制约只能取一些离散值,用以数学规划理论为基础的经典约束优化方法求解效果很差,用基于模拟退火算法的离散变量优化设计方法则可以方便快捷地获得满足各方面要求的最优设计方案。     

Optimization of truss bridges within a specified design domain using evolution strategies

This article reports and investigates the application of evolution strategies (ESs) to optimize the design of truss bridges. This is a challenging optimization problem associated with mixed design variables, since it involves identification of the bridge’s shape and topology configurations in addition to the sizing of the structural members for minimum weight. A solution algorithm to this problem is developed by combining different variable-wise versions of adaptive ESs under a common optimization routine. In this regard, size and shape optimizations are implemented using discrete and continuous ESs, respectively, while topology optimization is achieved through a discrete version coupled with a particular methodology for generating topological variations. In the study, a design domain approach is employed in conjunction with ESs to seek the optimal shape and topology configuration of a bridge in a large and flexible design space. It is shown that the resulting algorithm performs very well and produces improved results for the problems of interest.     

Evolutionary Algorithm Integrating Stress Heuristics for Truss Optimization

The optimal truss design using problem-oriented evolutionary algorithm is presented in the paper. The minimum weight structures subjected to stress and displacement constraints are searched. The discrete design variables are areas of members, selected from catalogues of available sections. The integration of the problem specific knowledge into the optimization procedure is proposed. The heuristic rules based on the concept of fully stressed design are introduced through special genetic operators, which use the information concerning the stress distribution of structural members. Moreover, approximated solutions obtained by deterministic, sequential discrete optimization methods are inserted into the initial population. The obtained hybrid evolutionary algorithm is specialized for truss design. Benchmark problems are calculated in numerical examples. The knowledge about the problem integrated into the evolutionary algorithm can enhance considerably the effectiveness of the approach and improve significantly the convergence rate and the quality of the results. The advantages and drawbacks of the proposed method are discussed.     

Adaptive multigrid method using duality in plane elasticity

This work presents an adaptive multigrid method for the mixed formulation of plane elasticity problems. First, a mixed‐hybrid formulation is introduced where the continuity of the normal components of the stress tensor is indirectly imposed using a Lagrange multiplier. Two different numerical approximations, naturally associated with the primal problem and the dual problem, are then proposed. The Complementary Energy Principle provides an a posteriori error estimate. For the effective solving of both systems of equations, a non‐standard multigrid algorithm has been designed that allows us to solve the two problems, dual and primal, with reasonable cost and in an integrated way. Finally, a significant numerical application is presented to check the efficiency of the error estimator and the good performance of the algorithm. Copyright © 2001 John Wiley & Sons, Ltd.     

Topology optimization of planar shape memory alloy thermal actuators using element connectivity parameterization

This paper presents the direct application of topology optimization to the design of shape memory alloy (SMA) thermal actuators. Because SMAs exhibit strongly nonlinear, temperature‐dependent material behavior, designing effective multidimensional SMA actuator structures is a challenging task. We pursue the use of topology optimization to address this problem. Conventional material scaling topology optimization approaches are hampered by the complexity of the SMA constitutive behavior combined with large actuator deflections. Therefore, for topology optimization we employ the element connectivity parameterization approach, which offers improved analysis convergence and robustness, as well as an unambiguous treatment of nonlinear materials. A path‐independent SMA constitutive model, aimed particularly at the NiTi R‐phase transformation, is employed, allowing efficient adjoint sensitivity analysis. The effectiveness of the proposed SMA topology optimization is demonstrated by numerical examples of constrained and unconstrained formulations of actuator stroke maximization, which provide insight into the characteristics of optimal SMA actuators. Copyright © 2011 John Wiley & Sons, Ltd.     

Detection of flaws in piezoelectric structures using extended FEM

An iterative method to treat the inverse problem of detecting cracks and voids in two‐dimensional piezoelectric structures is proposed. The method involves solving the forward problem for various flaw configurations, and at each iteration, the response of piezoelectric material is minimized at known specific points along the boundary to match measured data. Extended finite element method (XFEM) is employed for solving the forward problem as it allows the use of a single regular mesh for a large number of iterations with different flaw geometries. The minimization of cost function is performed by multilevel coordinate search (MCS) method. The algorithm is an intermediate between purely heuristic methods and methods that allow an assessment of the quality of the minimum obtained and is in spirit similar to the direct method for global optimization. In this paper, the XFEM‐MCS methodology is applied to two‐dimensional electromechanical problems where flaws considered are straight cracks and elliptical voids. The results show that this methodology can be effectively employed for damage detection in piezoelectric materials. Copyright © 2013 John Wiley & Sons, Ltd.