A Study on High School Mathematics Teaching Design Based on Teaching for Robust Understanding: Taking the Cosine Theorem as an Example

Teachers’teaching behavior plays a crucial role in students’development,and there are problems in the current teaching behavior of mathematics teachers such as ignoring students’cognitive needs,lack of equal opportunities for students’classroom performance as well as lack of formative evaluation of students.In order to solve the phenomenon,this paper analyzes and explains how to promote teaching based on the Teaching for Robust Understanding(TRU)evaluation framework with the goal of focusing on the development of all students,taking the teaching design of The Cosine Theorem as an example,and provides ideas and methods for first-line high school mathematics teachers.

PROCESSING PARAMETER OPTIMIZATION OF FDM BASED ON ROBUST DESIGN

The influence of processing parameters on the precision of parts fabricated by fused deposition modeling （FDM） technology is studied based on a series of performed experiments. Processing parameters of FDM in terms of wire-width compensation, extrusion velocity, filing velocity, and layer thickness are chosen as the control fac- tors. Robust design analysis and multi-index fuzzy comprehensive assessment method are used to obtain the opti- mal parameters. Results show that the influencing degrees of these four factors on the precision of as-processed parts are different. The optimizations of individual parameters and their combined effects are of the same impor- tance for a high precision manufacturing.

Reliability-based Robust Optimization Design of Automobile Components with Non-normal Distribution Parameters

In the reliability designing procedure of the vehicle components, when the distribution styles of the random variables are unknown or non-normal distribution, the result evaluated contains great error or even is wrong if the reliability value R is larger than 1 by using the existent method, in which case the formula is necessary to be revised. This is obviously inconvenient for programming. Combining reliability-based optimization theory, robust designing method and reliability based sensitivity analysis, a new method for reliability robust designing is proposed. Therefore the influence level of the designing parameters’ changing to the reliability of vehicle components can be obtained. The reliability sensitivity with respect to design parameters is viewed as a sub-objective function in the multi-objective optimization problem satisfying reliability constraints. Given the first four moments of basic random variables, a fourth-moment technique and the proposed optimization procedure can obtain reliability-based robust design of automobile components with non-normal distribution parameters accurately and quickly. By using the proposed method, the distribution style of the random parameters is relaxed. Therefore it is much closer to the actual reliability problems. The numerical examples indicate the following: (1) The reliability value obtained by the robust method proposed increases (】0.04%) comparing to the value obtained by the ordinary optimization algorithm; (2) The absolute value of reliability-based sensitivity decreases (】0.01%), and the robustness of the products’ quality is improved accordingly. Utilizing the reliability-based optimization and robust design method in the reliability designing procedure reduces the manufacture cost and provides the theoretical basis for the reliability and robust design of the vehicle components.

Design and Robust Performance Evaluation of a Fractional Order PID Controller Applied to a DC Motor

This paper proposes a methodology for the quantitative robustness evaluation of PID controllers employed in a DC motor. The robustness analysis is performed employing a 2~3 factorial experimental design for a fractional order proportional integral and derivative controller(FOPID), integer order proportional integral and derivative controller(IOPID)and the Skogestad internal model control controller(SIMC). The factors assumed in experiment are the presence of random noise,external disturbances in the system input and variable load. As output variables, the experimental design employs the system step response and the controller action. Practical implementation of FOPID and IOPID controllers uses the MATLAB stateflow toolbox and a NI data acquisition system. Results of the robustness analysis show that the FOPID controller has a better performance and robust stability against the experiment factors.

Multi-parameter Sensitivity Analysis and Application Research in the Robust Optimization Design for Complex Nonlinear System

The current research of complex nonlinear system robust optimization mainly focuses on the features of design parameters, such as probability density functions, boundary conditions, etc. After parameters study, high-dimensional curve or robust control design is used to find an accurate robust solution. However, there may exist complex interaction between parameters and practical engineering system. With the increase of the number of parameters, it is getting hard to determine high-dimensional curves and robust control methods, thus it＇s difficult to get the robust design solutions. In this paper, a method of global sensitivity analysis based on divided variables in groups is proposed. By making relevant variables in one group and keeping each other independent among sets of variables, global sensitivity analysis is conducted in grouped variables and the importance of parameters is evaluated by calculating the contribution value of each parameter to the total variance of system response. By ranking the importance of input parameters, relatively important parameters are chosen to conduct robust design analysis of the system. By applying this method to the robust optimization design of a real complex nonlinear system-a vehicle occupant restraint system with multi-parameter, good solution is gained and the response variance of the objective function is reduced to 0.01, which indicates that the robustness of the occupant restraint system is improved in a great degree and the method is effective and valuable for the robust design of complex nonlinear system. This research proposes a new method which can be used to obtain solutions for complex nonlinear system robust design.

Applying robust design to study the effects of stratigraphic characteristics on brittle failure and bump potential in a coal mine

Bumps and other types of dynamic failure have been a persistent, worldwide problem in the underground coal mining industry, spanning decades.For example, in just five states in the U.S.from 1983 to 2014,there were 388 reportable bumps.Despite significant advances in mine design tools and mining practices,these events continue to occur.Many conditions have been associated with bump potential, such as the presence of stiff units in the local geology.The effect of a stiff sandstone unit on the potential for coal bumps depends on the location of the stiff unit in the stratigraphic column, the relative stiffness and strength of other structural members, and stress concentrations caused by mining.This study describes the results of a robust design to consider the impact of different lithologic risk factors impacting dynamic failure risk.Because the inherent variability of stratigraphic characteristics in sedimentary formations,such as thickness, engineering material properties, and location, is significant and the number of influential parameters in determining a parametric study is large, it is impractical to consider every simulation case by varying each parameter individually.Therefore, to save time and honor the statistical distributions of the parameters, it is necessary to develop a robust design to collect sufficient sample data and develop a statistical analysis method to draw accurate conclusions from the collected data.In this study,orthogonal arrays, which were developed using the robust design, are used to define the combination of the(a) thickness of a stiff sandstone inserted on the top and bottom of a coal seam in a massive shale mine roof and floor,(b) location of the stiff sandstone inserted on the top and bottom of the coal seam,and(c) material properties of the stiff sandstone and contacts as interfaces using the 3-dimensional numerical model, FLAC3D.After completion of the numerical experiments, statistical and multivariate analysis are performed using the calculated results from the orthogonal arrays to analyze the effect of these variables.As a consequence, the impact of each of the parameters on the potential for bumps is quantitatively classified in terms of a normalized intensity of plastic dissipated energy.By multiple regression, the intensity of plastic dissipated energy and migration of the risk from the roof to the floor via the pillars is predicted based on the value of the variables.The results demonstrate and suggest a possible capability to predict the bump potential in a given rock mass adjacent to the underground excavations and pillars.Assessing the risk of bumps is important to preventing fatalities and injuries resulting from bumps.

ROBUST OPTIMUM DESIGN OF LAMINATED COMPOSITE PLATES

A last-ply failure (LPF) analysis method for laminated composite plates is incorpo- rated into the ?nite element code-ANSYS, and a robust optimum design method is presented. The composite structure is analyzed by considering both in-plane and out-of-plane loads. For a lamina, two major failure modes are considered: matrix failure and ?ber breakage that are characterized by the proper strength criteria in the literature. When a lamina has failed, the laminate sti?ness is modi?ed to re?ect the damage, and stresses in the structure are re-analyzed. This procedure is repeatedly performed until the whole structure fails and thus the ultimate strength is determined. A structural optimization problem is solved with the ?ber orientation and the lamina thickness as the design variables and the LPF load as the objective. Finally, the robust optimum design method for laminates is presented and discussed.

Adaptive robust dissipative designs on straight path control for underactuated ships

An adaptive robust control algorithm for ship straight path control system in the presence of both modeling uncertainties and the bounded disturbances is proposed. Motivated by the backstepping approach, the algorithm is developed by using the dissipation theory, such that the resulting dosed-loop system is both strictly dissipative and asymptotically adaptively stable for all admissible uncertainties. Also, it is able to steer an underactuated ship along a prescribed straight path with ultimate bounds under external disturbances induced by wave, wind and ocean current. When there are no disturbances, the straight path control can be implemented in a locally asymptotically stable manner. Simulation results on an ocean-going training ship ‘YULONG＇ are presented to validate the effectiveness of the algorithm.

CONSTRUCTION OF OPTIMAL BLOCKING SCHEMES FOR ROBUST PARAMETER DESIGNS

Robust parameter design （RPD） is an important issue in experimental designs. If all experimental runs cannot be performed under homogeneous conditions, blocking the units is effective. In this paper, we obtain the correspondence relation between fractional factorial RPDs and the blocking schemes for full factorial RPDs. In addition, we provide a construction of optimal blocking schemes that make all main effects and control-by-noise two-factor interactions estimable.

ROBUST DESIGN OF A TWO DEGREE FREEDOM SYSTEM.DUE TO THE PARAMETER UNCERTAINTY

A method based on the robust design optimization is presented to handle the structural uncertainty problems. The variations caused in dynamic performance can be expressed by the mean response and the standard deviation of the performance. The robust optimization approach, based on a multi-objective and non-deterministic method, attempts to both optimize the mean performance and minimize the variance of the performance simultaneously. The best possible design optimization is chosen by a trade-off decision. An example of robust design of a two degree freedom system is used to effectively illustrate the application in dynamics. The mass and stiffness uncertainty in the main system as well as the uncertainty of the mass, stiffness and damping in the absorber are considered all together in order to minimize the displacement response of the main system within a wide band of excitation frequencies. The robust optimization results show a significant improvement in performance compared with the conventional solution recommended from vibration textbooks. It is indicated that robust design methods have great potential for application in structural dynamics to deal with uncertainty problems.

MULTIDISCIPLINARY ROBUST OPTIMIZATION DESIGN

Because uncertainty factors inevitably exist under multidisciplinary designenvironment, a hierarchical multidisciplinary robust optimization design based on response surfaceis proposed. The method constructs optimization model of subsystem level and system level tocoordinate the coupling among subsystems, and also the response surface based on the artificialneural network is introduced to provide information for system level optimization tool to maintainthe independence of subsystems, i.e. to realize multidisciplinary parallel design. The applicationcase of electrical packaging demonstrates that reasonable robust optimum solution can be yielded andit is a potential and efficient multi-disciplinary robust optimization approach.

Robust Design Optimization Method for Centrifugal Impellers under Surface Roughness Uncertainties Due to Blade Fouling

Blade fouling has been proved to be a great threat to compressor performance in operating stage. The current researches on fouling-induced performance degradations of centrifugal compressors are based mainly on simplified roughness models without taking into account the realistic factors such as spatial non-uniformity and randomness of the fouling-induced surface roughness. Moreover, little attention has been paid to the robust design optimization of centrifugal compressor impellers with considerations of blade fouling. In this paper, a multi-objective robust design optimization method is developed for centrifugal impellers under surface roughness uncertainties due to blade fouling. A three-dimensional surface roughness map is proposed to describe the nonuniformity and randomness of realistic fouling accumulations on blades. To lower computational cost in robust design optimization, the support vector regression（SVR） metamodel is combined with the Monte Carlo simulation（MCS） method to conduct the uncertainty analysis of fouled impeller performance. The analyzed results show that the critical fouled region associated with impeller performance degradations lies at the leading edge of blade tip. The SVR metamodel has been proved to be an efficient and accurate means in the detection of impeller performance variations caused by roughness uncertainties. After design optimization, the robust optimal design is found to be more efficient and less sensitive to fouling uncertainties while maintaining good impeller performance in the clean condition. This research proposes a systematic design optimization method for centrifugal compressors with considerations of blade fouling, providing a practical guidance to the design of advanced centrifugal compressors.

Stamping Robust Design Method and Its Application

The application of the robust design to stamping is discussed in the paper. The regression equation of the quality index and its influencing factors can be attained by using method of central composite design. Stamping computer simulation takes the place of real stamping experiments to fulfill the special experimental conditions. The real stamping experiments prove the correctness and the feasibility of the stamping robust design method-central composite design.

Robust design of natural laminar flow supercritical airfoil by multi-objective evolution method

Abstract A transonic, high Reynolds number natural laminar flow airfoil is designed and studied. The γ-θ transition model is combined with the shear stress transport （SST） k-w turbulence model to predict the transition region for a laminar-turbulent boundary layer. The non-uniform free-form deformation （NFFD） method based on the non-uniform rational B-spline （NURBS） basis function is introduced to the airfoil parameterization. The non-dominated sorting genetic algorithm-II （NSGA-II） is used as the search algo- rithm, and the surrogate model based on the Kriging models is introduced to improve the efficiency of the optimization system. The optimization system is set up based on the above technologies, and the robust design about the uncertainty of the Mach number is carried out for NASA0412 airfoil. The optimized airfoil is analyzed and compared with the original airfoil. The results show that natural laminar flow can be achieved on a supercritical airfoil to improve the aerodynamic characteristic of airfoils.

Robust designs for models with possible bias and correlated errors

This paper studies the model-robust design problem for general models with an unknown bias or contamination and the correlated errors. The true response function is assumed to be from a reproducing kernel Hilbert space and the errors are fitted by the qth order moving average process MA（q）, especially the MA（1） errors and the MA（2） errors. In both situations, design criteria are derived in terms of the average expected quadratic loss for the least squares estimation by using a minimax method. A case is studied and the orthogonality of the criteria is proved for this special response. The robustness of the design criteria is discussed through several numerical examples.

Robust Trajectory and Communication Design for Angle-Constrained Multi-UAV Communications in the Presence of Jammers

This paper studies a multi-unmanned aerial vehicle(UAV)enabled wireless communication system,where multiple UAVs are employed to communicate with a group of ground terminals(GTs)in the presence of potential jammers.We aim to maximize the throughput overall GTs by jointly optimizing the UAVs’trajectory,the GTs’scheduling and power allocation.Unlike most prior studies,we consider the UAVs’turning and climbing angle constraints,the UAVs’three-dimensional(3D)trajectory constraints,minimum UAV-to-UAV(U2U)distance constraint,and the GTs’transmit power requirements.However,the formulated problem is a mixed-integer non-convex problem and is intractable to work it out with conventional optimization methods.To tackle this difficulty,we propose an efficient robust iterative algorithm to decompose the original problem be three sub-problems and acquire the suboptimal solution via utilizing the block coordinate descent(BCD)method,successive convex approximation(SCA)technique,and S-procedure.Extensive simulation results show that our proposed robust iterative algorithm offers a substantial gain in the system performance compared with the benchmark algorithms.

MODELING AND ROBUST DESIGN OF REMANUFACTURING LOGISTICS NETWORKS BASED ON DESIGN OF EXPERIMENT

The uncertainty of time, quantity and quality of recycling products leads tothe bad stability and flexibility of remanufacturing logistics networks, and general design onlycovered the minimizing logistics cost, thus, robust design is presented here to solve theuncertainty. The mathematical model of remanufacturing logistics networks is built based onstochastic distribution of uncontrollable factors, and robust objectives are presented. Theintegration of mathematical simulation and design of experiment method is performed to do sensitiveanalysis. The influence of each factor and level on the system is investigated, and the main factorsand optimum combination are studied. The numbers of factors, level of each factor and designprocess of experiment are investigated as well. Finally, the process of robust design based ondesign of experiment is demonstrated by a detailed example.

Robust Optimum Design of Thrust Hydrodynamic Bearings for Hard Disk Drives

This paper describes the robust optimum design which combines the geometrical optimization method proposed by Hashimoto and statistical method. Recently, 2.5″ hard disk drives (HDDs) are widely used for mobile devices such as laptops, video cameras and car navigation systems. In mobile applications, high durability towards external vibrations and shocks are essentials to the bearings of HDD spindle motor. In addition, the bearing characteristics are influenced by manufacturing error because of small size of the bearings of HDD. In this paper, the geometrical optimization is carried out to maximize the bearing stiffness using sequential quadratic programming to improve vibration characteristics. Additionally, the bearing stiffness is analyzed considering dimensional tolerance of the bearing using statistical method. The dimensional tolerance is assumed to distribute according to the Gaussian distribution, and then the bearing stiffness is estimated by combining the expectation and standard deviation. As a result, in the robust optimum design, new groove geometry of bearing can be obtained in which the bearing stiffness is four times higher than the stiffness of conventional spiral groove bearing. Moreover, the bearing has lower variability compared with the result of optimum design neglecting dimensional tolerance.

A novel robust design method for improving stability of optimized structures

It is known that structural optimization may lead to designs of structures having low stability and sometimes even kinematically unstable designs. This paper presents a robust design method for improving the stability of opti mized structures. A new approach is proposed, in which cer tain perturbation loads are introduced and the corresponding compliance is added to the objective function as a penaliza tion. The stability of the optimized structures can thus be improved substantially by considering structural responses to the original and the introduced loads. Numerical exam ples show the simplicity and effectiveness of the proposed method.

Non-probabilistic Robust Optimal Design Method

For the purpose of dealing with uncertainty factors in engineering optimization problems, this paper presents a new non-probabilistic robust optimal design method based on maximum variation estimation. The method analyzes the effect of uncertain factors to objective and constraints functions, and then the maximal variations to a solution are calculated. In order to guarantee robust feasibility the maximal variations of constraints are added to original constraints as penalty term; the maximal variation of objective function is taken as a robust index to a solution; linear physical programming is used to adjust the values of quality characteristic and quality variation, and then a bi-level mathematical robust optimal model is constructed. The method does not require presumed probability distribution of uncertain factors or continuous and differentiable of objective and constraints functions. To demonstrate the proposed method, the design of the two-bar structure acted by concentrated load is presented. In the example the robustness of the normal stress, feasibility of the total volume and the buckling stress are studied. The robust optimal design results show that in the condition of maintaining feasibility robustness, the proposed approach can obtain a robust solution which the designer is satisfied with the value of objective function and its variation.