The development and application of the Gauss nonlinearly constrained optimization method

Presented in this paper is a new optimization technique, called the Gauss nonlinearly constrained technique, which is applicable to design problems with nonlinear objective functions and constraints. The technique is an extension of a previously developed method for linear constraints, referred to as the Gauss constrained technique. Both of these techniques, based on the Gauss unconstrained method, have been developed so that the Kuhn-Tucker conditions are automatically satisfied when the procedure terminates.     

A performance index for topology and shape optimization of plate bending problems with displacement constraints

This paper presents a performance index for topology and shape optimization of plate bending problems with displacement constraints. The performance index is developed based on the scaling design approach. This performance index is used in the Performance-Based Optimization (PBO) method for plates in bending to keep track of the performance history when inefficient material is gradually removed from the design and to identify optimal topologies and shapes from the optimization process. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates with single and multiple displacement constraints under various loading conditions. The topology optimization and shape optimization are undertaken for the same plate in bending, and the results are evaluated by using the performance index. The proposed performance index is also employed to compare the efficiency of topologies and shapes produced by different optimization methods. It is demonstrated that the performance index developed is an effective indicator of material efficiency for bending plates. From the manufacturing and efficient point of view, the shape optimization technique is recommended for the optimization of plates in bending. Received November 27, 1998?Revised version received June 6, 1999     

Explicit level set and density methods for topology optimization with equivalent minimum length scale constraints

The goal of this paper is to introduce local length scale control in an explicit level set method for topology optimization. The level set function is parametrized explicitly by filtering a set of nodal optimization variables. The extended finite element method (XFEM) is used to represent the non-conforming material interface on a fixed mesh of the design domain. In this framework, a minimum length scale is imposed by adopting geometric constraints that have been recently proposed for density-based topology optimization with projections filters. Besides providing local length scale control, the advantages of the modified constraints are twofold. First, the constraints provide a computationally inexpensive solution for the instabilities which often appear in level set XFEM topology optimization. Second, utilizing the same geometric constraints in both the density-based topology optimization and the level set optimization enables to perform a more unbiased comparison between both methods. These different features are illustrated in a number of well-known benchmark problems for topology optimization.

     

Truss structure optimization using adaptive multi-population differential evolution

This paper applies multi-population differential evolution (MPDE) with a penalty-based, self-adaptive strategy—the adaptive multi-population differential evolution (AMPDE)—to solve truss optimization problems with design constraints. The self-adaptive strategy developed in this study is a new adaptive approach that adjusts the control parameters of MPDE by monitoring the number of infeasible solutions generated during the evolution process. Multiple different minimum weight optimization problems of the truss structure subjected to allowable stress, deflection, and kinematic stability constraints are used to demonstrate that the proposed algorithm is an efficient approach to finding the best solution for truss optimization problems. The optimum designs obtained by AMPDE are better than those found in the current literature for problems that do not violate the design constraints. We also show that self-adaptive strategy can improve the performance of MPDE in constrained truss optimization problems, especially in the case of simultaneous optimization of the size, topology, and shape of truss structures.     

Experience with approximate reliability-based optimization methods II: an exhaust system problem

Traditional reliability-based design optimization (RBDO) requires a double-loop iteration process. The inner optimization loop is to find the reliability and the outer is the regular optimization loop to optimize the RBDO problem with reliability objectives or constraints. It is known that the computation can be prohibitive when the associated function evaluation is expensive. This situation is even worse when a large number of reliability constraints are present. As a result, many approximate RBDO methods, which convert the double loop to a single loop, have been developed. In this research, an engineering problem with a large number of constraints (144) is designed to test RBDO methods based on the first-order reliability method (FORM), including single- and double-loop methods. In addition to the number of constraints, this problem possesses many local minimums. Some original authors of the RBDO methods are also asked to solve the same problem. The results and the efficiencies for different methods are published and discussed.     

A modified Covariance Matrix Adaptation Evolution Strategy with adaptive penalty function and restart for constrained optimization

In the last decades, a number of novel meta-heuristics and hybrid algorithms have been proposed to solve a great variety of optimization problems. Among these, constrained optimization problems are considered of particular interest in applications from many different domains. The presence of multiple constraints can make optimization problems particularly hard to solve, thus imposing the use of specific techniques to handle fitness landscapes which generally show complex properties. In this paper, we introduce a modified Covariance Matrix Adaptation Evolution Strategy (CMA-ES) specifically designed for solving constrained optimization problems. The proposed method makes use of the restart mechanism typical of most modern variants of CMA-ES, and handles constraints by means of an adaptive penalty function. This novel CMA-ES scheme presents competitive results on a broad set of benchmark functions and engineering problems, outperforming most state-of-the-art algorithms as for both efficiency and constraint handling.     

Design of structure/control systems with transient response constraints exhibiting relative minima

Structural optimization problems involving dynamic behaviour constraints often exhibit nonconvex design spaces. The direct application of a global optimization algorithm requires a large number of function evaluations which in turn require a large number of dynamic structural analyses. This work presents a strategy aimed at finding the global optimum for problems with transient dynamic behaviour constraints based on approximation concepts. The method consists of generating and solving a sequence of approximate problems using a global optimizer. The approximations are explicit and capture most of the inherent nonconvexity of the exact functions. A simple example. problem is presented to illustrate the procedure set forth.     

Matrix-free algorithm for the optimization of multidisciplinary systems

Multidisciplinary engineering systems are usually modeled by coupling software components that were developed for each discipline independently. The use of disparate solvers complicates the optimization of multidisciplinary systems and has been a long-standing motivation for optimization architectures that support modularity. The individual discipline feasible (IDF) formulation is particularly attractive in this respect. IDF achieves modularity by introducing optimization variables and constraints that effectively decouple the disciplinary solvers during each optimization iteration. Unfortunately, the number of variables and constraints can be significant, and the IDF constraint Jacobian required by most conventional optimization algorithms is prohibitively expensive to compute. Furthermore, limited-memory quasi-Newton approximations, commonly used for large-scale problems, exhibit linear convergence rates that can struggle with the large number of design variables introduced by the IDF formulation. In this work, we show that these challenges can be overcome using a reduced-space inexact-Newton-Krylov algorithm. The proposed algorithm avoids the need for the explicit constraint Jacobian and Hessian by using a Krylov iterative method to solve the Newton steps. The Krylov method requires matrix-vector products, which can be evaluated in a matrix-free manner using second-order adjoints. The Krylov method also needs to be preconditioned, and a key contribution of this work is a novel and effective preconditioner that is based on approximating a monolithic solution of the (linearized) multidisciplinary system. We demonstrate the efficacy of the algorithm by comparing it with the popular multidisciplinary feasible formulation on two test problems.     

Stress-constrained topology optimization with design-dependent loading

The purpose of this paper is to apply stress constraints to structural topology optimization problems with design-dependent loading. A comparison of mass-constrained compliance minimization solutions and stress-constrained mass minimization solutions is also provided. Although design-dependent loading has been the subject of previous research, only compliance minimization has been studied. Stress-constrained mass minimization problems are solved in this paper, and the results are compared with those of compliance minimization problems for the same geometries and loading. A stress-relaxation technique is used to avoid the singularity in the stress constraints, and these constraints are aggregated in blocks to reduce the total number of constraints in the optimization problem. The results show that these design-dependent loading problems may converge to a local minimum when the stress constraints are enforced. The use of a continuation method where the stress-constraint aggregation parameter is gradually increased typically leads to better convergence; however, this may not always be possible. The results also show that the topologies of compliance-minimization and stress-constrained solutions are usually vastly different, and the sizing optimization of a compliance solution may not lead to an optimum.     

A hybrid CBO–PSO algorithm for optimal design of truss structures with dynamic constraints

The vibration domain of structures can be reduced by imposing some constraints on their natural frequencies. For this purpose optimal design of structures under frequency constraints is required which involves highly non-linear and non-convex problems. In this paper an efficient hybrid algorithm is developed for solving such optimization problems. This algorithm utilizes the recently developed colliding bodies optimization (CBO) algorithm as the main engine and uses the positive properties of the particle swarm optimization (PSO) algorithm to increase the efficiency of the CBO. The distinct feature of the present hybrid algorithm is that it requires no parameter tuning. The CBO is known for being parameter independent, and avoiding the use of the traditional penalty method to handle the constraints upholds this property. Two mathematical constrained functions taken from the literature are studied to verify the performance of the algorithm. The algorithm is then applied to optimize truss structures with frequency limitations. The numerical results demonstrate the efficiency of the presented algorithm for this class of problems.     

Optimum design of stiffened cylindrical shells with natural frequency constraints

The design optimization of axially loaded, simply supported stiffened cylindrical shells for minimum mass is considered. The design variables are thickness of shell wall, thicknesses and depths of rings and stringers, number/spacing of rings and stringers. Natural frequency, local and overall buckling strengths and direct stress constraints are considered in the design problems. Three different combinations of stiffeners are considered. In each case, the independent effects of behaviour constraints are also studied. The optimum designs are achieved with one of the standard nonlinear constrained optimization techniques (Davidon-Fletcher-Powell method with interior penalty function formulation) and few optimal solutions are checked for the satisfaction of Kuhn-Tucker conditions.     

A new optimality criteria method for shape optimization of natural frequency problems

A new shape optimization method for natural frequency problems is presented. The approach is based on an optimality criterion for general continuum solids, which is derived in this paper for the maximization of the first natural frequency with a volume constraint. An efficient redesign rule for frequency problems is developed to achieve the required shape modifications. The optimality criterion is extended to volume minimization problems with multiple frequency constraints. The nonparametric geometry representation creates a complete design space for the optimization problem, which includes all possible solutions for the finite element discretization. The combination with the optimality criteria approach results in a robust and fast convergence, which is independent of the number of design variables. Sensitivity information of objective function and constraints are not required, which allows to solve the structural analysis task using fast and reliable industry standard finite element solvers like ABAQUS, ANSYS, I-DEAS, MARC, NASTRAN, or PERMAS. The new approach is currently being implemented in the optimization system TOSCA.     

Optimal design of hybrid laminated composite plates with dynamic and static constraints

Optimization procedures are presented that consider the static and dynamic characteristic constraints for laminated composite plates and hybrid laminated composite plates subject to a concentrated load on the center of the plate. The design variables adopted are ply angle or ply thickness. Considered constraints are deflection, natural frequency and specific damping capacity. Using a recursive linear programming method, nonlinear optimization problems are solved, and by introducing the design scaling factor, the number of iterations is reduced significantly. Relating interactive optimization procedures with the finite element method analysis, various hybrid composite plates with arbitrary boundary conditions can be designed optimally. In the optimization procedure, verification of analysis and design of the laminated composite plates are compared with a previous paper. Various design results are presented on laminated composite plates and hybrid laminated composite plates.     

Exponential penalty function formulation for multilevel optimization using the analytical target cascading framework

An exponential penalty function (EPF) formulation based on method of multipliers is presented for solving multilevel optimization problems within the framework of analytical target cascading. The original all-at-once constrained optimization problem is decomposed into a hierarchical system with consistency constraints enforcing the target-response coupling in the connected elements. The objective function is combined with the consistency constraints in each element to formulate an augmented Lagrangian with EPF. The EPF formulation is implemented using double-loop (EPF I) and single-loop (EPF II) coordination strategies and two penalty-parameter-updating schemes. Four benchmark problems representing nonlinear convex and non-convex optimization problems with different number of design variables and design constraints are used to evaluate the computational characteristics of the proposed approaches. The same problems are also solved using four other approaches suggested in the literature, and the overall computational efficiency characteristics are compared and discussed.     

A new approach for the solution of singular optima in truss topology optimization with stress and local buckling constraints

The present paper investigates problems of truss topology optimization under local buckling constraints. A new approach for the solution of singular problems caused by stress and local buckling constraints is proposed. At first, a second order smooth-extended technique is used to make the disjoint feasible domains connect, then the so-called ε-relaxed method is applied to eliminate the singular optima from problem formulation. By means of this approach, the singular optimum of the original problem caused by stress and local buckling constraints can be searched approximately by employing the algorithms developed for sizing optimization problems with high accuracy. Therefore, the numerical problem resulting from stress and local buckling constraints can be solved in an elegant way. The applications of the proposed approach and its effectiveness are illustrated with several numerical examples. Received May 2, 2000     

An application of dynamic modeling to the sea shore rotation planning problem in the navy

The navy sea shore rotation planning problem is presented as multicriteria dynamic network optimization problem with side constraints. Using the simplex primal optimization code of Professor J. Kennington and a specially designed heuristic rounding routine we developed a computer model to run practical problems with different input parameters.     

Sizing and layout design of truss structures under dynamic and static constraints with an integrated particle swarm optimization algorithm

Natural frequencies offer useful knowledge on the dynamic response of the structures. It is possible to avoid from the destructive effects of dynamic loads on the structures by optimizing layout and size of their subject to constraints on natural frequencies. Since optimization problems including frequency constraints are highly nonlinear, this kind of problems forms a challenging area to test the performance of the different optimization techniques. This study tests the performance of an integrated particle swarm optimization algorithm (iPSO), a new particle swarm optimizer integrated with the improved fly-back mechanism and the weighted particle concept, in four weight minimization of truss structures with sizing and layout variables under multiple frequency constraints. Optimization results demonstrate that the new algorithm is competitive with other state-of-the-art metaheuristic algorithms in dynamic and static structural optimization problems.     

Optimal design of truss structures for size and shape with frequency constraints using a collaborative optimization strategy

A new metaheuristic strategy is proposed for size and shape optimization problems with frequency constraints. These optimization problems are considered to be highly non-linear and non-convex. The proposed strategy extends the idea of using a single optimization process to a series of collaborative optimization processes. In this study, a modified teaching-learning-based optimization (TLBO), which is a relatively simple algorithm with no intrinsic parameters controlling its performance, is utilized in a collaborative framework and introduced as a higher-level TLBO algorithm called school-based optimization (SBO). SBO considers a school with multiple independent classrooms and multiple teachers with inter-classroom collaboration where teachers are reassigned to classrooms based on their fitness. SBO significantly improves the both exploration and exploitation capabilities of TLBO without increasing the algorithm's complexity. In addition, since the SBO algorithm uses multiple independent classrooms with interchanging teachers, the algorithm is less likely to be influenced by local optima. A parametric study is conducted to investigate the effects of the number of classes and the class size, which are the only parameters of SBO. The SBO algorithm is applied to five benchmark truss optimization problems with frequency constraints and the statistical results are compared to other optimization techniques in the literature. The quality and robustness of the results indicate the efficiency of the proposed SBO algorithm.     

Balancing stochastic two-sided assembly line with multiple constraints using hybrid teaching-learning-based optimization algorithm

Two-sided assembly lines are usually found in the factories which produce large-sized products. In most literatures, the task times are assumed to be deterministic while these tasks may have varying operation times in real application, causing the reduction of performance or even the infeasibility of the schedule. Moreover, the ignorance of some specific constraints including positional constraints, zoning constraints and synchronism constraints will result in the invalidation of the schedule. To solve this stochastic two-sided assembly line balancing problem with multiple constraints, we propose a hybrid teaching-learning-based optimization (HTLBO) approach which combines both a novel teaching-learning-based optimization algorithm for global search and a variable neighborhood search with seven neighborhood operators for local search. Especially, a new priority-based decoding approach is developed to ensure that the selected tasks satisfy most of the constraints identified by multiple thresholds of the priority value and to reduce the idle times related to sequence-dependence among tasks. Experimental results on benchmark problems demonstrate both remarkable efficiency and universality of the developed decoding approach, and the comparison among 11 algorithms shows the effectiveness of the proposed HTLBO.