Truss optimization with natural frequency constraints using generalized normal distribution optimization

The newly proposed Generalized Normal Distribution Optimization (GNDO) algorithm is used to design the truss structures with optimal weight. All trusses optimized have frequency constraints, which make them very challenging optimization problems. A large number of locally optimal solutions and non-convexity of search space make them difficult to solve, therefore, they are suitable for testing the performance of optimization algorithm. This work investigates whether the proposed algorithm is capable of coping with such problems. To evaluate the GNDO algorithm, three benchmark truss optimization problems are considered with frequency constraints. Numerical data show GNDO’s reliability, stability, and efficiency for structural optimization problems than other meta-heuristic algorithms. We thoroughly analyse and investigate the performance of GNDO in this engineering area for the first time in the literature.

     

Design optimization with static and dynamic displacement constraints

A design optimization procedure using a sequential linear programming technique is proposed in this paper to design minimum weight structures subjected to frequency response and static displacement constraints. The merit of the proposed approach is that the reanalyses of the static and dynamic responses, as well as the computations of the static and dynamic sensitivity data, are performed in a reduced approximate model. A significant saving of computer time for large scale structures is expected. Two numerical examples show good results of this method.     

2D shape optimization with static and dynamic constraints

The paper presents an approach that allows us to consider in the shape optimization several static loading conditions together with constraints imposed on eigenfrequencies. The idea of the method is based upon simultaneous solutions of equations and inequalities arising from the Kuhn-Tucker necessary conditions for an optimum problem. The paper is illustrated with four examples in which stress and eigenfrequency are active constraints.     

Difficulties in truss topology optimization with stress,local buckling and system stability constraints

A serlous difficulty in topology optimization with only stress andlocal buckling constraints was pointed out recently by Zhou (1996a). Possibilities for avoiding this pitfall are (i) inclusion of system stability constraints and (ii) application of imperfections in the ground structure. However, it is shown in this study that the above modified procedures may also lead to erroneous solutions which cannot be avoided without changing the ground structure.     

Application of optimization methods to helicopter rotor blade design

The minimum weight design of helicopter rotor blades with constraints on multiple coupled flap-lag natural frequencies is studied in this paper. A constraint is also imposed on the minimum value of the autorotational inertia of the blade to ensure sufficient autorotational inertia to autorotate in case of an engine failure. A stress constraint is used to guard against structural failure due to blade centrifugal forces. Design variables include blade taper ratio, dimensions of the box beam located inside the airfoil and magnitudes of the nonstructural weights. The program CAMRAD is used for the blade modal analysis and the program CONMIN for the optimization. In addition, a linear approximation involving Taylor series expansion is used to reduce the analysis effort. The procedure contains a sensitivity analysis which consists of analytical derivatives of the objective function, the autorotational inertia constraint and the stress constraints. A central finite difference scheme is used for the derivatives of the frequency constraints. Optimum designs are obtained for both rectangular and tapered blades. The paper also discusses the effect of adding constraints on higher frequencies and stresses on the optimum designs. b box beam width - c chord - f ₁,f ₃,f ₄ first three lead-lag dominated frequencies (elastic modes) - f ₂,f ₅ first two flapping dominated frequencies (elastic modes) - g constraint function - h box beam height - h(z) box beam height variation along blade span - n number of blades - r _j distance from the root to the center of thej-th segment - t ₁,t ₂,t ₃ box beam wall thicknesses - x, y, z reference axes - A box beam cross-sectional area - AI autorotational inertia - E Young's modulus - F objective function - FS factor of safety - GJ torsional stiffness - I _x ,I _y total principal area moments of inertia about reference axes - L _j length ofj-th segment - M _j total mass ofj-th segment - N total number of blade segments - NDV number of design variables - R blade radius - W total blade weight - W() blade weight as a function of design variable - W _b box beam weight - W _o nonstructural blade weight (weight of skin, honeycomb, etc. along with tuning/lumped weights) - prescribed autorotational inertia - design variable increment - _h taper ratio inz direction - _i i-th design variable - _j mass density of thej-th segment - _j stress inj-th segment - _max maximum allowable stress - blade RPM - r root value - t tip value - L lower bound - U upper bound - ^ approximate value This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States     

Bridge topology optimisation with stress, displacement and frequency constraints

The principal stress based evolutionary structural optimisation method is presented herein for topology optimisation of arch, tied arch, cable-stayed and suspension bridges with both stress and displacement constraints. Two performance index formulas are developed to determine the efficiency of the topology design. A refined mesh scheme is proposed to improve the details of the final topology without resorting to the complete analysis of a finer mesh. Furthermore, cable-supported bridges are optimised with frequency constraint incorporating the “nibbling” technique. The applicability, simplicity and effectiveness of the method are demonstrated through the topology optimisation of the four types of bridges.     

Models and methods for frequency assignment with cumulative interference constraints

In this paper, a realistic modeling of interferences for frequency assignment in hertzian telecommunication networks is presented. In contrast with traditional interference models based only on binary interference constraints involving two frequencies, this new approach considers the case of cumulative disruptions that are modeled through a unique non‐binary constraint. To deal with these complex constraints, we propose extensions of classical integer linear programming formulations. On a set of realistic instances, we propose hybrid constraint programming and large neighborhood search solution methods to solve minimum interference and minimum span frequency assignment problems. We compare their performances with those of existing heuristics. Finally, we show how the end‐user benefits from using the cumulative model instead of the traditional one.     

Exporting water wave optimization concepts to modified simulated annealing algorithm for size optimization of truss structures with natural frequency constraints

This paper proposes an improved version of a recently proposed modified simulated annealing algorithm (MSAA) named as an improved MSAA (I-MSAA) to tackle the size optimization of truss structures with frequency constraint. This kind of problem is problematic because its feasible region is non-convex while the boundaries are highly non-linear. The main motivation is to improve the exploitative behavior of MSAA, taking concept from water wave optimization metaheuristic (WWO). An interesting concept of WWO is its breaking operation. Thirty functions extracted from the CEC2014 test suite and four benchmark truss optimization problems with frequency constraints are explored for the validity of the proposed algorithm. Numerical results indicate that I-MSAA is more reliable, stable and efficient than those found by other existing metaheuristics in the literature.

     

An optimality criterion method for structures with stress,displacement and frequency constraints

Presented in this paper is a procedure, based on optimality criterion methods, for the minimum weight design of structures subjected to stress, displacement and natural frequency constraints. The technique presented is a combination of a previously developed method for stress and displacement constraints alone and one for frequency limited structures. The method is applied to a delta-wing example, and the optimal designs obtained are compared to previously published results. The new method is capable of obtaining the optimal design in a small number of iterations, without significant calculations beyond a standard analysis. No approximate analyses or determination of large numbers of Lagrange multipliers are involved.     

Topology optimization for minimum weight with compliance and stress constraints

The paper deals with a formulation for the topology optimization of elastic structures that aims at minimizing the structural weight subject to compliance and local stress constraints. The global constraint provides the expected stiffness to the optimal design while a selected set of local enforcements require feasibility with respect to the assigned strength of material. The Drucker?CPrager failure criterion is implemented to handle materials with either equal or unequal behavior in tension and compression. A suitable relaxation of the equivalent stress measure is implemented to overcome the difficulties related to the singularity problem. Numerical examples are presented to discuss the features of the achieved optimal designs along with performances of the adopted procedure. Comparisons with pure compliance?Cbased or pure stress?Cbased strategies are also provided to point out differences arising in the optimal design with respect to conventional approaches, depending on the assumed material behavior.     

Fast structural optimization with frequency constraints by genetic algorithm using adaptive eigenvalue reanalysis methods

Structural optimization with frequency constraints is highly nonlinear dynamic optimization problems. Genetic algorithm (GA) has greater advantage in global optimization for nonlinear problem than optimality criteria and mathematical programming methods, but it needs more computational time and numerous eigenvalue reanalysis. To speed up the design process, an adaptive eigenvalue reanalysis method for GA-based structural optimization is presented. This reanalysis technique is derived primarily on the Kirsch’s combined approximations method, which is also highly accurate for case of repeated eigenvalues problem. The required number of basis vectors at every generation is adaptively determined and the rules for selecting initial number of basis vectors are given. Numerical examples of truss design are presented to validate the reanalysis-based frequency optimization. The results demonstrate that the adaptive eigenvalue reanalysis affects very slightly the accuracy of the optimal solutions and significantly reduces the computational time involved in the design process of large-scale structures.     

Weld optimization with stress constraints and thermal load

We consider weld optimization with stress constraints and thermal load. Our interest for weld optimization is motivated by the development of exhaust aftertreatment systems, because most designs of these systems involve components that are assembled with welds and the designs are exposed to heat from the exhaust. We find that it is possible to optimize the welds with commercial software. An example is given in the end of the paper.     

Optical flow estimation: an error analysis of gradient-based methods with local optimization

Multiple views of a scene can provide important information about the structure and dynamic behavior of three-dimensional objects. Many of the methods that recover this information require the determination of optical flow-the velocity, on the image, of visible points on object surfaces. An important class of techniques for estimating optical flow depend on the relationship between the gradients of image brightness. While gradient-based methods have been widely studied, little attention has been paid to accuracy and reliability of the approach. Gradient-based methods are sensitive to conditions commonly encountered in real imagery. Highly textured surfaces, large areas of constant brightness, motion boundaries, and depth discontinuities can all be troublesome for gradient-based methods. Fortunately, these problematic areas are usually localized can be identified in the image. In this paper we examine the sources of errors for gradient-based techniques that locally solve for optical flow. These methods assume that optical flow is constant in a small neighborhood. The consequence of violating in this assumption is examined. The causes of measurement errors and the determinants of the conditioning of the solution system are also considered. By understanding how errors arise, we are able to define the inherent limitations of the technique, obtain estimates of the accuracy of computed values, enhance the performance of the technique, and demonstrate the informative value of some types of error.     

Interior point methods for power flow optimization with security constraints

This paper deals with power flow optimization with security constraints, focusing on the problem of short‐term hydroelectric scheduling, called predispatch. Since the energy demand varies throughout the day, the generation must satisfy daily targets, established by long‐term scheduling models. This study considers that the hydroelectric plants and transmission systems must provide an optimal flow of energy under security constraints that allow meeting energy demands for normal operating conditions and when disturbances happen. Algebraic techniques are used to exploit the sparse structure of the problem, targeting the design of an interior point algorithm, efficient in terms of robustness and computational time. Case studies compare the proposed approach with a general purpose optimization solver for quadratic problems and an algorithm for the predispatch problem that does not consider security constraints. The results show the benefits of using the method proposed in the paper, obtaining optimal power flow that is suitable to consider contingencies, with numerical stability and appropriate computational time.     

Structural modification with frequency response constraints for undamped mdof systems

A structural modification method which aims to change the structural design to match the desired frequency response behaviors is introduced. The calculations of frequency response sensitivities and reanalysis of frequency responses of a modified structure are efficiently carried out in a reduced modal space. Either a least square solution or an optimal solution with minimum structural weight subjected to various types of frequency response constraints can be obtained through an iteration loop which repeatedly uses the frequency response sensitivity data and reanalysis technique to achieve the design goal. A tremendous cost saving is expected when this is implemented in practical engineering design.     

Hypotrochoid spiral optimization approach for sizing and layout optimization of truss structures with multiple frequency constraints

Engineering with Computers - The primary aim of this article is to present a new improved version of the spiral optimization algorithm (SPO) for shape and size optimization of truss structures...     

Truss optimization with multiple frequency constraints and automatic member grouping

This paper deals with sizing and shape structural optimization problems with respect to the minimization of the masses of truss structures considering multiple natural frequencies as the constraints of the problems. The sizing and shape design variables are discrete and continuous, respectively. It can be attractive to use a reduced number of distinct cross-sectional areas minimizing costs of fabrication, transportation, storing, checking, welding, and so on. Also, it is expected a labor-saving when the structure is welded, checked and so on. On the other hand, one can observe that the task of discovering the optimum member grouping is not trivial and leads to an exhaustive trial-and-error process. Cardinality constraints are adopted in order to obtain an automatic variable linking searching for the best member grouping of the bars of the trusses analyzed in this paper. A CRPSO (Craziness based Particle Swarm Optimization) is the search algorithm adopted in this paper. This algorithm uses a modified velocity expression and an operator called “craziness velocity” in order to avoid premature convergence. An Adaptive Penalty Method is adopted to handle the constraints. Six truss structures are analyzed, presenting very interesting results providing curves of tradeoff between the optimized weights versus the number of distinct cross-sectional areas used in these solutions.     

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.     

Evolutionary topology optimization of continuum structures with stress constraints

Structural and Multidisciplinary Optimization - In this work, we propose to extend the bi-directional evolutionary structural optimization (BESO) method for compliance minimization design subject...     

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.