Heuristic Methods in Discrete Structural Optimization

Multi-objective optimization design of pile-anchor structures for slopes based on reliability theory considering the spatial variability of soil properties

In order to save project investment and reduce project quantity, the optimal design of aqueduct is the main problem in the project therefore, carrying out the optimal design and simulation analysis of U-shaped thin-walled aqueduct has important engineering application value. The paper takes Sha River aqueduct as the research object and adopts the direct search and point checking method of mixed discrete variable optimisation design. It also adopts the structural optimisation design method based on genetic algorithm and gives the optimal design scheme of U-shaped aqueduct. The optimisation design method has the advantages of fast convergence speed, high convergence efficiency and strong global search ability. The results show that, U-shape aqueduct structure is smooth, it has good mechanical properties, and the structural stress is small and can meet the strength requirement. The maximum vertical displacement of the aqueduct occurs in the middle of the span, the vertical displacement value is small, which can meet the requirement of stiffness.

IntroductionIn recent years, SMPMSM is quietly famous compared with other conventional motors in industrial applications due to higher efficiency, higher reliability, low cost, and smooth torque [1]–[3]. In general, several objective optimizations need to be consider, which required an optimization technique for an electrical motor. This paper deals with an optimal design of a surface-mounted permanent magnet synchronous machine (SMPMSM) with an exact analytical subdomain model by using a genetic algorithm method. To analyze the characteristic of permanent magnet (PM) motors, the classical optimization method, such as the finite element method (FEM), is intensively used. However, FEM has several time problems that require a longer computational time to evaluate the performance of PM motors. This problem can be overcome by using a genetic algorithm (GA) method combined with a subdomain model (SD), which developed an improved performance of SMPMSM, for instance, total harmonic distortion (THDv) and phase back-EMF. In this design, a three-phase 12-slot/8-pole PM motor is established with an exact SD model with radial magnetization patterns. Then, the GA ensemble with SD model to search the optimality of SMPMSM machine design. In the final analysis, the optimal new design of SMPMSM demonstrated by comparing with the initial design that is investigated by FEM. The result of induced back-EMF, and magnetic flux density of optimal design is compared with the initial design to show the advantages of GA optimization method.Analytical Model, Genetic Algorithm and Parameter of optimizationThe purpose of this analysis is to investigate the optimization design of a three-phase 12-slot/8-pole SMPMSM with a radial magnetization pattern. This optimization problem has been solved by obtains the minimum THDv as a fitness function. In order to process the optimization design of SMPMSM, several parameters need to be optimized i.e magnet pole-pitch, αp and slot-opening angle, boa. The roulette method is used to select the objective function of optimization with 50 maximum generations per iteration. The cross and mutation probability need to be set which requires linear and uniform recombination which is 0.89 and 0.025 respectively. The parameters of 12-slot/8-pole SMPM motor as follows: radius of the rotor, Rr = 23mm, the radius of PM surface Rm = 26mm, the inner radius of the stator Rsb = 42.5mm, axial length, la = 50mm, air-gap length, lg = 1mm, remanence of the magnet, Br, 1.12T, relative coil permeability, µr = 1.05 with the number of pole pairs, Np is 8 and number of stator slots, Ns is 12.The result, Discussion and ConclusionThe phase back electromotive force has a significant relationship with flux-linkage because phase back electromotive force is derived from the rate of change of flux-linkage. In Fig. 1. show the waveform of phase back electromotive force against rotor position of initial and optimal design in radial magnetization patterns. The initial design of phase back electromotive force waveform shows more trapezoidal compare with the optimal design which has a sinusoidal waveform. The comparison of total harmonic distortion between the initial and optimal design of SMPMSM are demonstrated in Fig. 2 below. The result from Fig.2. shows the highest total harmonic distortion of 12-slot/8-pole motor is an initial design which is 11.34% compared with the optimal design where 1.3%. The magnitude of harmonic in phase back electromotive force |En| of initial design shows 22.24 V compare to optimal design which is 17.4 V. This shows a good agreement that the optimal motor design gives lowest total harmonic distortion compare with the initial design. The optimizes of SMPMSM motor with the following parameters are given in Table 1. **

Recently, methods for computing D-optimal designs for population pharmacokinetic studies have become available. However there are few publications that have prospectively evaluated the benefits of D-optimality in population or single-subject settings. This study compared a population optimal design with an empirical design for estimating the base pharmacokinetic model for enoxaparin in a stratified randomized setting. The population pharmacokinetic D-optimal design for enoxaparin was estimated using the PFIM function (MATLAB version 6.0.0.88). The optimal design was based on a one-compartment model with lognormal between subject variability and proportional residual variability and consisted of a single design with three sampling windows (0-30 min, 1.5-5 hr and 11-12 hr post-dose) for all patients. The empirical design consisted of three sample time windows per patient from a total of nine windows that collectively represented the entire dose interval. Each patient was assigned to have one blood sample taken from three different windows. Windows for blood sampling times were also provided for the optimal design. Ninety six patients were recruited into the study who were currently receiving enoxaparin therapy. Patients were randomly assigned to either the optimal or empirical sampling design, stratified for body mass index. The exact times of blood samples and doses were recorded. Analysis was undertaken using NONMEM (version 5). The empirical design supported a one compartment linear model with additive residual error, while the optimal design supported a two compartment linear model with additive residual error as did the model derived from the full data set. A posterior predictive check was performed where the models arising from the empirical and optimal designs were used to predict into the full data set. This revealed the "optimal" design derived model was superior to the empirical design model in terms of precision and was similar to the model developed from the full dataset. This study suggests optimal design techniques may be useful, even when the optimized design was based on a model that was misspecified in terms of the structural and statistical models and when the implementation of the optimal designed study deviated from the nominal design.

Considering uncertainties in engineering optimum design is often a requirement. Here, the use of the deterministic optimum design as the reference point in g-dominance is proposed. The multiobjective optimum robust design in a structural engineering test case where uncertainties in the external loads are taken into account is proposed as application, where the simultaneous minimization of the constrained weight average and the standard deviation of the constraints violation are the objective functions. Results include a comparison between both non-dominated sorting genetic algorithm II (NSGA-II) and strength Pareto evolutionary algorithm (SPEA2), including S-metric (hypervolume) statistical comparisons with and without the g-dominance approach. The methodology is capable to provide robust optimum structural frame designs successfully.

This paper proposes a hybrid heuristic and criteria-based method of optimum design which combines the advantages of both the iterated simulated annealing (SA) algorithm and the rigorously derived optimality criteria (OC) for structural optimum design of reinforced concrete (RC) buildings under multi-load cases based on the current Chinese design codes. The entire optimum design procedure is divided into two parts: strength optimum design and stiffness optimum design. A modified SA with the strategy of adaptive feasible region is proposed to perform the discrete optimization of RC frame structures under the strength constraints. The optimum stiffness design is conducted using OC method with the optimum results of strength optimum design as the lower bounds of member size. The proposed method is integrated into the commercial software packages for building structural design, SATWE, and for finite element analysis, ANSYS, for practical applications. Finally, two practical frame-shear-wall structures (15-story and 30-story) are optimized to illustrate the effectiveness and practicality of the proposed optimum design method.

This chapter introduces the improvement and application of swarm algorithm named Group Search Optimizer (GSO) in civil structure optimization design. GSO is a new and robust stochastic searching optimizer, as it is based on PS (produce and scrounger) model and the animal scanning mechanisms. An improved group search optimizer named IGSO was presented based on harmony search mechanism and GSO. The implementation of IGSO for different optimal purposes is presented in detail, including the application of IGSO to truss structure shape optimal design, to truss structure dynamic optimal design, to truss structure topology optimization design. Different truss structures are used to test the GSO and IGSO in structural shape optimization, dynamic optimization and topology optimization.

A generalized 2-DOF model for optimal design of MDOF structures controlled by Tuned Mass Damper Inerter (TMDI)

This paper presents the generalized compound scaling algorithm and its application to optimum weight design of plate structures. The optimum designs are reached by simply scaling the design variables to an optimum intersection of multiple constraints. A four-noded isoparametric plate element is used for modeling the structure. Sensitivity computations and the optimization algorithm are discussed. The optimization cost involved (excluding the finite element analyses) is very small using this algorithm. The procedure is demonstrated on three example problems using stress and displacement constraints with side bounds on the design variables.

This work integrates developments in simplified dynamic modeling of unanchored fluid-filled tanks, stochastic modeling of earthquake loading time histories, probabilistic structural analysis tools, cost analyses, and multi-criteria optimal design formulations for the design optimization of unanchored fluid-filled cylindrical tanks operating in an uncertain seismic environment. An appropriate simplified mechanical model is developed which includes effects from the sloshing at the liquid free surface, the soil flexibility and the separation of the base plate of the tank from its foundation, which occurs during strong ground motion. A class of stochastic processes with frequency content and time-varying intensity characteristics determined by seismological parameters is used to model the earthquake loading time history. The reliability of the cylindrical tank against failure of the shell structure is computed using latest developments in efficient Monte Carlo simulation methods. A multi-criteria design optimization methodology is then used to obtain the optimal design characteristics of the tank system that meet cost and reliability constrains. Issues related to reliability estimation and optimization are addressed. The methodology is useful for optimally designing fluid-filled structures that operate in a seismic environment to withstand earthquakes with minimal social and economical losses during the lifetime of the structure.

Finite-element approximation of 2D elliptic optimal design

<p>This paper defines basic operations of neutrosophic numbers and neutrosophic number functions for objective functions and constraints in optimization models. Then, we propose a general neutrosophic number optimization model for the optimal design of truss structures. The application and effectiveness of the neutrosophic number optimization method are demonstrated through the design example of a two-bar truss structure under indeterminate environment to achieve the minimum weight objective under stress and stability constraints. The comparison of the neutrosophic number optimal design method with traditional optimal design methods proves the usability and suitability of the presented neutrosophic number optimization design method under an indeterminate / neutrosophic number environment.</p>

This paper presents a heuristic design optimization method specifically developed for practicing structural engineers. Practical design optimization problems are often governed by buildability constraints. The majority of optimization methods that have recently been proposed for design optimization under buildability constraints are based on evolutionary computing. While these methods are generally easy to implement, they require a large number of function evaluations (finite element analyses), and they involve algorithmic parameters that require careful tuning. As a consequence, both the computation time and the engineering time are high. The discrete design optimization algorithm presented in this paper is based on the optimality criteria method for continuous optimization. It is faster than an evolutionary algorithm and it is free of tuning parameters. The algorithm is successfully applied to two classical benchmark problems (the design of a ten-bar truss and an eight-story frame) and to a practical truss design optimization problem.

In this chapter, the optimum design problem of steel space frames is formulated according to the provisions of LRFD-AISC (Load and Resistance Factor Design-American Institute of Steel Corporation). The weight of the steel frame is taken as the objective function to be minimized. The design optimization problem necessitates selection of steel sections for the members of the steel frame from the available steel profiles lists. This turns the design optimization problem into discrete programming problem. Obtaining the optimum solution of such programming problems is cumbersome with mathematical programming techniques. On the other hand with the use of recently developed metaheuristic techniques that are based on swarm intelligence, the solution of the same problem becomes straightforward. Five different structural optimization algorithms are developed which are based on ant colony optimization, particle swarm optimizer, artificial bee colony algorithm, firefly algorithm, and cuckoo search algorithm, respectively. Two real size steel space frames; one rigidly connected and the other pin jointed are designed using each of these algorithms. The optimum designs obtained by these techniques are compared and performance of each version is evaluated. It is noticed that most of swarm intelligence-based algorithms are simple and robust techniques that determine the optimum solution of structural design optimization problems efficiently without requiring much of a mathematical struggle.

Drive-coils and armature are the pivotal assembly in synchronous induction coil launchers (SICL). In order to improve the electrical-to-kinetic energy conversion efficiency of SICL, optimum structural design of drive-coils and armature are done. The model of optimum design is put forward with regarding the electrical-to-kinetic energy conversion efficiency as the target function based on the electro-mechanical model of SICL. Taking a SICL with 60 millimeter caliber for example, the axial length, the radial thickness and turns of drive-coils, the axial length and the radial thickness of armature are optimally analyzed. The relation between these parameters and the electrical-to-kinetic energy conversion efficiency of SICL is gained, and the optimum design alternative is put forward in this paper at last.