Hammersley Sequence Research Articles

An improved adaptive sampling strategy for freeform surface inspection on CMM

The components with freeform surface are widely used in industrial fields. The machining quality of freeform surface becomes increasingly significant. Coordinate measuring machine (CMM), as a conventional dimensional measuring instrument, is commonly used to inspect freeform surface. The sampling strategy, consisting of distribution of sampling points and sample size, is important to the effectiveness and accuracy of the measurements performed on CMM. The present work aims at proposing an improved adaptive sampling strategy based on a machining error model (MEM) for freeform surface inspection on CMM. The machining error model is built to determine the distribution of sampling points adaptively. In addition, Hammersley sequence is adopted to address the aliasing problem. Experiments are performed to compare presented strategy with four well-known sampling strategies. Results prove that the MEM is valid and the present strategy is reliable and effective.

Multi-objective optimization of the hybrid wind/solar/fuel cell distributed generation system using Hammersley Sequence Sampling

As the development of China's economy, environmental problems in China become more and more serious. Solar energy and wind energy are considered as ones of the best choices to solve the environmental problems in China and the hybrid wind/solar distributed generation (DG) system has received increasing attention recently. However, the instability and intermittency of the wind and solar energy throw a huge challenge on designing of the hybrid system. In order to ensure the continuous and stable power supply, optimal unit sizing of the hybrid wind/solar DG system should be taken into consideration in the design of the hybrid system. This paper establishes a multi-objective optimization framework based on cost, electricity efficiency and energy supply reliability models of the hybrid DG system, which is composed of wind, solar and fuel cell generation systems. Detailed models of each unit for the hybrid wind/solar/fuel cell system were established. Advanced ε-constraints method based on Hammersley Sequence Sampling was employed in the multi-objective optimization of the hybrid DG system. The approximate Pareto surface of the multi-objective optimization problems with a range of possible design solutions and a logical procedure for searching the global optimum solution for decision makers were presented. In this way, this work provided an efficient method for decision makers in the design of the hybrid wind/solar/fuel cell system.

Response surface method for assessing energy production from geopressured geothermal reservoirs

Developing low-enthalpy geothermal resources along the US Gulf Coast is attractive for reducing global warming and providing clean energy. In this work, synthetic yet representative models for typical geopressured geothermal reservoirs located along the US Gulf Coast are considered. A Box–Behnken experimental design is used to select a small set of these models to perform detailed reservoir simulation runs. Full quadratic linear models are fit to the simulation results, and their sufficiency is confirmed by comparing them to kriging response surfaces. To achieve a higher degree of efficiency in using the response surfaces, Hammersley sequence sampling (HSS) method is used instead of traditional Monte Carlo sampling. HSS ensures that the factor space is sampled more uniformly and the response distribution is converged in less time. By evaluating these proxy models in the sampled factor space, the sensitivity and uncertainty of the response to the factors can be assessed. In this work, the sensitivity and uncertainty of engineered convection is assessed. For quantifying engineered convection, five uncertain reservoir attributes were selected. The response was defined as the net extracted enthalpy. In particular, two different designs for harvesting energy from geothermal reservoirs were compared using the response surfaces. In the modeled systems, results show that the regular design is more effective than the reverse design for extracting energy from geopressured geothermal reservoirs.

Comparison of Monte Carlo and quasi-Monte Carlo technique in structure and relaxing dynamics of polymer in dilute solution

Structure and dynamics of polymer in solvent solution is an important area of research since the functional properties of polymer are largely dependent on the morphology of the polymers in solution. This structure related properties are especially important in case of surface science where the phase-separated morphology in the micro/nano scale dictates the properties of the product. Modeling polymers in solution is an efficient way to determine the morphology and thus the properties of the products. It saves time as well as helps to design novel materials with desired properties. Polymers in solution systems are generally modeled with bead spring model and Monte Carlo or importance sampling Monte Carlo simulations is used to find the optimal configuration where the energy of the system is minimized. Often in these simulations, random numbers are used in the Monte Carlo steps. Normally random numbers try to form clusters and do not cover the entire dimension of the system. Thus the minimum energy structures obtained from simulations with random numbers are not optimal configuration of the system. In the present work a lattice-based model is used for polymer solution system and importance sampling Monte Carlo is used for simulation. Quasi-random numbers generated from Hammersley sequence sampling (HSS) are used in the simulation steps for stochastic selection polymers and its movements. Quasi-random numbers obtained from HSS are random in nature and they have n-dimensional uniformity. They do not form clusters and the structural configuration obtained using quasi-random numbers are optimal in nature. The optimal configurations of the polymers as obtained from random number and quasi-random number are compared. The result shows that simulation with HSS attains a lower energy state after initial quench. At the late stage of spinodal decomposition, the structure factor decrease-showing Ostwald ripening which is not observed from simulation with random numbers.

Efficient ant colony optimization for computer aided molecular design: Case study solvent selection problem

In this paper, we propose a novel computer-aided molecular design (CAMD) methodology for the design of optimal solvents based on an efficient ant colony optimization (EACO) algorithm. The molecular design problem is formulated as a mixed integer nonlinear programming (MINLP) model in which a solvent performance measure is maximized (solute distribution coefficient) subject to structural feasibility, property, and process constraints. In developing the EACO algorithm, the better uniformity property of Hammersley sequence sampling (HSS) is exploited. The capabilities of the proposed methodology are illustrated using a real world case study for the design of an optimal solvent for extraction of acetic acid from waste process stream using liquid–liquid extraction. The UNIFAC model based on the infinite dilution activity coefficient is used to estimate the mixture properties. New solvents with better targeted properties are proposed.

Efficient Ant Colony Optimization (EACO) Algorithm for Deterministic Optimization

In this paper, an efficient ant colony optimization (EACO) algorithm is proposed based on efficient sampling method for solving combinatorial, continuous and mixed-variable optimization problems. In EACO algorithm, Hammersley Sequence Sampling (HSS) is introduced to initialize the solution archive and to generate multidimensional random numbers. The capabilities of the proposed algorithm are illustrated through 9 benchmark problems. The results of the benchmark problems from EACO algorithm and the conventional ACO algorithm are compared. More than 99% of the results from the EACO show efficiency improvement and the computational efficiency improvement range from 3% to 71%. Thus, this new algorithm can be a useful tool for large-scale and wide range of optimization problems. Moreover, the performance of the EACO is also tested using the five variants of ant algorithms for combinatorial problems.

Kriging Methodology for Surrogate-Based Airfoil Shape Optimization

In this paper, a computational fluid dynamics optimization strategy-based surrogate model, which is used for the predictions of vertical aerodynamic force (Cl), is proposed. An airfoil shape optimization problem has been formulated, and the particle swarm optimization algorithm technique is used to solve the problem along with the inclusion of constructed surrogate model. This is used to replace the actual CFD algorithms. Ordinary kriging and Hammersley sequence sampling approach are applied to construct and enhance the effectiveness of the constructed surrogate model, respectively. The geometry of the airfoils (NACA 2411 and 0012) is described with the help of parametric section method, and the flow around the airfoils is solved by using the Panel method. The constructed surrogate model is more accurate and limits their estimation error within the order of 10−2–10−6. The computational time required to solve the formulated ASO problem, which will take 630 s while using a normal optimization scheme in a computer system which has 2GB of DDR3 RAM, 2.0GHz of processor speed, and 1MB of L2 cache memory, is drastically reduced to 56.7 s while using the surrogate-based optimization, and also, the optimized airfoils have improved Cl when compared to the original airfoils. The optimized airfoils were validated using experimental data.

Sampling Strategy for Free-Form Surface Inspection Using Coordinate Measuring Machines<sup></sup>

Due to the complexity and non-rotational symmetry of free-form surface, it is difficult to achieve accurate and efficient inspection method. In order to solve this problem, three types of sampling sequences are proposed to specify a set of measuring points of free-form surface. For comparing the results of different sampling strategies, the profile errors of free-form surface are calculated based on a quasi particle swarm optimization (QPSO) searching the transformation parameters to implement localization and surface subdivision method finding the closest points on the design model corresponding to measured points. In order to obtain effective sampling strategies, four design models are generated by non-uniform rational basis spline (NURBS) and parts are manufactured on two machining centers to obtain surfaces of different roughness and measured on CMMs by selecting different sampling methods and sample sizes. The profile errors of parts are calculated by the proposed method and CMMs software, respectively. The results show that randomized Hammersley sampling sequence and medium sample size are preferred for the profile error inspection of given parts if accuracy and time are all considered. The research provides a method for free-form surface accurate inspection while minimizing the sampling time and cost.

Robust multi-objective optimization design of TMD control device to reduce tall building responses against earthquake excitations using genetic algorithms

Tuned Mass Dampers (TMDs) are a well-accepted control device widely used by the civil engineering community. The main purpose of this study is the robust multi-objective optimization design of this device using Genetic Algorithms (GAs) to control the structural vibrations against earthquakes. To enhance the performance of the TMD system, its parameters, including mass, stiffness, and damping ratio, have been optimally designed using multi-objective genetic algorithms. For doing this, three non-commensurable objective functions, namely: maximum displacement, maximum velocity, and maximum acceleration of each floor, are considered, which are to be minimized simultaneously. For this purpose, a fast and elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) approach is used to find a set of Pareto-optimal solutions. Moreover, in order to take into account the uncertainties existing in the system, a robust design optimization procedure is performed using the Hammersley sequence sampling approach. In this study, the example building is modeled as a 3-D frame, and its responses are evaluated using coupled multi-mode analysis. From the numerical results of the study, it is found that the robust TMD system is capable of providing a reduction of about 28% on maximum displacement of the building.

Adaptive Hybrid Surrogate Modeling for Complex Systems

This paper explores the effectiveness of the recently developed surrogate modeling method, the Adaptive Hybrid Functions, when applied in conjunction with different sampling techniques and different sample sizes. The Adaptive Hybrid Functions is a hybrid surrogate modeling method that seeks to exploit the advantages of each component surrogate. To explore its effectiveness, the Adaptive Hybrid Functions is applied to model four disparate complex engineering systems, namely: 1) wind farm power generation, 2) product platform planning (for universal electric motors), 3) three-pane window heat transfer, and 4) onshore wind farm cost estimation. The effectiveness of the following three distinct sampling techniques is investigated: 1) Latin hypercube sampling, 2) Sobol’s quasi-random sequence, and 3) the Hammersley sequence sampling. Cross-validation is used to evaluate the accuracy of the resulting surrogate models. It was observed that the Sobol’s sequence and the Latin hypercube sampling techniques provided better accuracy in the case of high-dimensional problems, whereas the Hammersley sequence sampling technique performed better in the case of low-dimensional problems. It was further observed that a monotonic increase in modeling accuracy is not necessarily accomplished by simply increasing the sample size for different problems. Overall, the Adaptive Hybrid Functions provided acceptable to high accuracy in representing complex system behavior.

Development of Artificial Neural Network Based Metamodels for Inactivation of Anthrax Spores in Ventilated Spaces Using Computational Fluid Dynamics

Linear, quadratic, and artificial neural network (ANN)-based metamodels were developed for predicting the extent of anthrax spore inactivation by chlorine dioxide in a ventilated three-dimensional space over time from computational fluid dynamics model (CFD) simulation data. Dimensionless groups were developed to define the design space of the problem scenario. The Hammersley sequence sampling (HSS) method was used to determine the sampling points for the numerical experiments within the design space. A CFD model, comprised of multiple submodels, was applied to conduct the numerical experiments. Large eddy simulation (LES) with the Smagorinsky subgridscale model was applied to compute the airflow. Anthrax spores were modeled as a dispersed solid phase using the Lagrangian treatment. The disinfectant transport was calculated by solving a mass transport equation. Kinetic decay constants were included for spontaneous decay of the disinfectant and for the reaction of the disinfectant with the surfaces of the three-dimensional space. To enhance the mixing of the disinfectant with the room air, a momentum source was included in the simulation. An inactivation rate equation accounted for the reaction between the spores and the disinfectant. The ANN-based metamodels were most successful in predicting the number of viable bioaerosols remaining in an arbitrary enclosed space. Sensitivity analysis showed that the mass fraction of the disinfectant, inactivation rate constant, and contact time had the most influence on the inactivation of the spores.

Development of high performance catalysts for CO oxidation using data-based modeling

This paper presents a model-aided approach to the development of catalysts for CO oxidation. This is in contrast to the traditional methodology whereby experiments are guided based on experience and intuition of chemists. The proposed approach operates in two stages. To screen a promising combination of active phase, promoter and support material, a powerful “space-filling” experimental design (specifically, Hammersley sequence sampling) was adopted. The screening stage identified Au–ZnO/Al 2O 3 as a promising recipe for further optimization. In the second stage, the loadings of Au and ZnO were adjusted to optimize the conversion of CO through the integration of a Gaussian process regression (GPR) model and the technique of maximizing expected improvement. Considering that Au constitutes the main cost of the catalyst, we further attempted to reduce the loading of Au with the aid of GPR, while keeping the low-temperature conversion to a high level. Finally we obtained 2.3%Au–5.0%ZnO/Al 2O 3 with 21 experiments. Infrared reflection absorption spectroscopy and hydrogen temperature-programmed reduction confirmed that ZnO significantly promotes the catalytic activity of Au.

Development of metamodels for predicting aerosol dispersion in ventilated spaces

Abstract Artificial neural network (ANN) based metamodels were developed to describe the relationship between the design variables and their effects on the dispersion of aerosols in a ventilated space. A Hammersley sequence sampling (HSS) technique was employed to efficiently explore the multi-parameter design space and to build numerical simulation scenarios. A detailed computational fluid dynamics (CFD) model was applied to simulate these scenarios. The results derived from the CFD simulations were used to train and test the metamodels. Feed forward ANN’s were developed to map the relationship between the inputs and the outputs. The predictive ability of the neural network based metamodels was compared to linear and quadratic metamodels also derived from the same CFD simulation results. The ANN based metamodel performed well in predicting the independent data sets including data generated at the boundaries. Sensitivity analysis showed that particle tracking time to residence time and the location of input and output with relation to the height of the room had more impact than the other dimensionless groups on particle behavior.

Uncertainty propagation for effective reduced-order model generation

This work describes a procedure to quantify effective uncertainty propagation through the development of a reduced-order model. To accomplish this objective, the concept of a random fuzzy variable is applied to represent both random and systematic errors associated with uncertain variables. A procedure to obtain feasible combinations of multiple uncertain variables is described. To predict the output probabilistic measure accurately with a minimum number of sample, efficient sampling that combines the techniques of Latin hypercube sampling and Hammersley sequence pairing is used. Based on the output data a reduced-order model is generated using the well known Karhunen-Loève expansion. The results show that the outputs of the reduced-order model track the outputs of the nonlinear physics-based model satisfactorily. A chemical reactor is used to demonstrate the concepts.

Modified genetic algorithm with sampling techniques for chemical engineering optimization

In this work, we develop a new efficient technique to enhance the optimization ability, and to improve the convergence speed of genetic optimization algorithm. We investigate and introduce a number of sampling techniques to generate a good set of initial population that encourages the exploration through out the search space and hence achieves better discovery of possible global optimum in the solution space. The introduced sampling techniques include Latin hypercube sampling (LHS), Faure sequence sampling (FSS), and Hammersley sequence sampling (HSS). The performances of the proposed algorithms and a conventional genetic algorithm using uniformly random population are compared, both in terms of solution quality and speed of convergence. A number of test problems and a case study, optimization of multi-effect distillation, demonstrate the feasibility and effectiveness of the proposed techniques. With the same parameters, our technique provides a better solution and converge to the global optimum faster than the traditional genetic algorithm.

Process Yield Improvement Through Optimum Design of Fixture Layouts in 3D Multistation Assembly Systems

Fixtures control the positions and orientations of parts in an assembly process. Inaccuracies of fixture locators or nonoptimal fixture layouts can result in the deviation of a workpiece from its design nominal and lead to overall product dimensional variability and low process yield. Major challenges involving the design of a set of fixture layouts for multistation assembly system can be enumerated into three categories: (1) high-dimensional design space since a large number of locators are involved in the multistation system, (2) large and complex design space for each locator since the design space represents the area of a particular part or subassembly surfaces on which a locator is placed, (here, the design space varies with a particular part design and is further expanded when parts are assembled into subassemblies), and (3) the nonlinear relations between locator nominal positions and key product characteristics. This paper presents a new approach to improve process yield by determining an optimum set of fixture layouts for a given multistation assembly system, which can satisfy (1) the part and subassembly locating stability in each fixture layout and (2) the fixture system robustness against environmental noises in order to minimize product dimensional variability. The proposed methodology is based on a two-step optimization which involves the integration of genetic algorithm and Hammersley sequence sampling. First, genetic algorithm is used for design space reduction by estimating the areas of optimal fixture locations in initial design spaces. Then, Hammersley sequence sampling uniformly samples the candidate sets of fixture layouts from those predetermined areas for the optimum. The process yield and part instability index are design objectives in evaluating candidate sets of fixture layouts. An industrial case study illustrates and validates the proposed methodology.

Selection of sampling points for accurate evaluation of flatness error using coordinate measuring machine

Dimensional inspection, in integrated manufacturing environments, requires accurate inspection while minimizing the cost and time of inspection. The selection of sampling plan—sample size and sample point locations, the method of evaluating the form error and the nature of the manufactured surfaces will play an important role in deciding the best inspection strategy to be adopted. This paper deals with the strategy for evaluation of flatness error which is one of the most commonly used form tolerances for control of manufactured surfaces. Investigations have been carried out to ascertain the influence of surface quality (surface roughness) in determining the sampling strategy for accurate determination of flatness error while inspecting on a coordinate measuring machine (CMM). The sampling plan utilizes the Hammersley sequence for point location and the flatness error is evaluated using the minimum zone method (MZM) based on computational geometry techniques. Results indicate that the surface roughness influences the accuracy of inspection and can be used as a parameter for determining an initial sample size for the determination of flatness error.

Hammersley Sampling and Support-Vector-Regression-Driven Launch Vehicle Design

Multidisciplinary design optimization of solid-fueled multistage space launch vehicles requires complex highdimensional simulation models. To improve on the ability to efficiently explore and approximate large subspaces of these models, this research develops a new set of experimental designs for metamodeling with support vector regression. We propose to Latinize and improve the orthogonality of Hammersley sequence sampling for creating nearly orthogonal and excellent space-filling designs. Multiple measures are used to assess the quality of designed experiments. The designs are used to create metamodels of trajectory simulation of space launch vehicles using support vector regression. A hybrid genetic algorithm uses metamodels to identify minimum-launch-weight space launch vehicles. This approach resulted in an overall rapid and efficient design optimization scheme.

New stochastic simulation capability applied to the GREET model

In 1995, the Center for Transportation Research (CTR) of Argonne National Laboratory (ANL) began to develop a model, called GREET (Greenhouse gases, Regulated Emissions, and Energy use in Transportation), for estimating the full fuel-cycle energy and emissions impacts of alternative transportation fuels and advanced vehicle technologies. The parametric assumptions used in the GREET model involve uncertainties. A new stochastic simulation tool, developed by Vishwamitra Research Institute (VRI), is built into the GREET model to address uncertainties. This paper presents the methodology and features of this new stochastic simulation tool and evaluates the performance of the sampling techniques in the tool. The new tool is interfaced through the graphical user interface (GUI) to perform the stochastic simulation. In general, five steps need to be followed to run a complete simulation: 1) Specify probability distribution functions; 2) Indicate the number of samples and the sampling technique; 3) Define the forecast variables; 4) Delete distribution functions (if necessary); and 5) Propagate the uncertainties and statistically analyze the outputs. The GREET model contains more than 700 default distribution functions for a wide variety of key parameters and as many as 3000 forecast variables. The stochastic simulation tool has been developed to incorporate 11 probability distribution function types for representing uncertain parameters and four sampling techniques (Monte Carlo sampling [MCS], Hammersley Sequence sampling [HSS], Latin Hypercube sampling [LHS] and Latin Hypercube Hammersley sampling [LHHS]) for stochastic simulation. To evaluate the performance of the four sampling techniques, 16 independent stochastic simulation runs were conducted in GREET and the output results were analyzed and compared. With the same number of samples, the output distribution curve simulated by HSS is the smoothest corresponding to the highest level of uniformity. To achieve the same level of smoothness as HSS with 1,000 samples, LHHS needs to be simulated with ∼1500 samples and LHS and MCS with ∼3,000 samples. As a result, HSS can achieve more than 200% reduction in running time compared to LHS or MCS without compromising the accuracy and quality of the prediction curves. The simulated mean values are close enough to the actual mean value (within ±1%) despite the selection of sampling technique and the number of samples (between 1,000 and 4,000). The standard deviation values from each other are close enough as well (within ±5%). It shows the trend that the increasing number of samples makes the simulated mean value marginally closer to the actual mean value; however, the improvement effect is negligible. The simulation time is strictly positive-correlated with the number of samples; therefore, the trade-off between extending simulation time and improving the smoothness of the output distribution curve needs to be carefully assessed. A new stochastic simulation tool has been developed to be built into Argonne’s GREET model to enhance its capability for addressing uncertainty. This new tool guides the user in each step of the process through the user-friendly GUI windows. According to the performance comparison among the four sampling techniques, HSS was found to be the most efficient technique. Therefore, HSS was set as the default technique in GREET.

Efficient molecular simulations for environmentally benign processes

An efficient sampling technique namely the Hammersley sequence sampling (HSS) technique based on quasi-random numbers is applied to molecular simulations to increase computational efficiency and reduce the error in property calculations for efficient and environmentally friendly product and process design. Due to its k-dimensional uniformity properties, this sampling technique reduces the computational intensity and converges faster as compared to conventional Monte Carlo (MC) method based on pseudo-random numbers. Two case studies are presented in order to show the effect of the sampling technique on molecular simulations. The first case study involves the computation of equation of state (EOS) for a Lennard Jones fluid and the second case study concerns the prediction of Gibbs free energies of transfer for the calculation of octanol–water partition coefficients. The results show that the HSS technique provides about 3-fold gain in efficiency and reduces the number of cycles required to reach equilibration. Furthermore, the errors in property calculations are reduced for both of the case studies.