Multi-Degree Reduction of Bézier Curves with Distance and Energy Optimization

In this paper, we propose a new approach to the problem of degree reduction of Bézier curves based on the given endpoint constraints. A differential term is added for the purpose of controlling the smoothness to a certain extent. Considering the adjustment of second derivative in curve design, a modified objective function including two parts is constructed here. One part is a kind of measure of the distance between original high order Bézier curve and degree-reduced curve. The other part represents the second derivative of degree-reduced curve. We tackle two kinds of conditions which are position vector constraint and tangent vector constraint respectively. The explicit representations of unknown points are presented. Some examples are illustrated to show the influence of the differential terms to approximation and smoothness effect.

Constrained multi-degree reduction of triangular Bézier surfaces

This paper proposes and applies a method to sort two-dimensional control points of triangular Bezier surfaces in a row vector. Using the property of bivariate Jacobi basis functions, it further presents two algorithms for multi-degree reduction of triangular Bezier surfaces with constraints, providing explicit degree-reduced surfaces. The first algorithm can obtain the explicit representation of the optimal degree-reduced surfaces and the approximating error in both boundary curve constraints and corner constraints. But it has to solve the inversion of a matrix whose degree is related with the original surface. The second algorithm entails no matrix inversion to bring about computational instability, gives stable degree-reduced surfaces quickly, and presents the error bound. In the end, the paper proves the efficiency of the two algorithms through examples and error analysis.

Constrained multi-degree reduction of rational Bézier curves using reparameterization

Applying homogeneous coordinates, we extend a newly appeared algorithm of best constrained multi-degree reduc- tion for polynomial Bézier curves to the algorithms of constrained multi-degree reduction for rational Bézier curves. The idea is introducing two criteria, variance criterion and ratio criterion, for reparameterization of rational Bézier curves, which are used to make uniform the weights of the rational Bézier curves as accordant as possible, and then do multi-degree reduction for each component in homogeneous coordinates. Compared with the two traditional algorithms of "cancelling the best linear common divisor" and "shifted Chebyshev polynomial", the two new algorithms presented here using reparameterization have advantages of simplicity and fast computing, being able to preserve high degrees continuity at the end points of the curves, do multi-degree reduction at one time, and have good approximating effect.

Multi-degree reduction of tensor product Bézier surfaces with conditions of corners interpolations

This paper studies the multi-degree reduction of tensor product B(?)zier surfaces with any degree interpolation conditions of four corners, which is urgently to be resolved in many CAD/CAM systems. For the given conditions of corners interpolation, this paper presents one intuitive method of degree reduction of parametric surfaces. Another new approximation algorithm of multi-degree reduction is also presented with the degree elevation of surfaces and the Chebyshev polynomial approximation theory. It obtains the good approximate effect and the boundaries of degree reduced surface can preserve the prescribed continuities. The degree reduction error of the latter algorithm is much smaller than that of the first algorithm. The error bounds of degree reduction of two algorithms are also presented .

Multi-degree reduction of NURBS curves based on their explicit matrix representation and polynomial approximation theory

NURBS curve is one of the most commonly used tools in CAD systems and geometric modeling for its various specialties, which means that its shape is locally adjustable as well as its continuity order, and it can represent a conic curve precisely. But how to do degree reduction of NURBS curves in a fast and efficient way still remains a puzzling problem. By applying the theory of the best uniform approximation of Chebyshev polynomials and the explicit matrix representation of NURBS curves, this paper gives the necessary and sufficient condition for degree reducible NURBS curves in an explicit form. And a new way of doing degree reduction of NURBS curves is also presented, including the multi-degree reduction of a NURBS curve on each knot span and the multi-degree reduction of a whole NURBS curve. This method is easy to carry out, and only involves simple calculations. It provides a new way of doing degree reduction of NURBS curves, which can be widely used in computer graphics and industrial design.

Optimal constrained multi-degree reduction of Bézier curves with explicit expressions based on divide and conquer

We decompose the problem of the optimal multi-degree reduction of Bézier curves with corners constraint into two simpler subproblems, namely making high order interpolations at the two endpoints without degree reduction, and doing optimal degree reduction without making high order interpolations at the two endpoints. Further, we convert the second subproblem into multi-degree reduction of Jacobi polynomials. Then, we can easily derive the optimal solution using orthonormality of Jacobi polynomials and the least square method of unequally accurate measurement. This method of 'divide and conquer' has several advantages including maintaining high continuity at the two endpoints of the curve, doing multi-degree reduction only once, using explicit approximation expressions, estimating error in advance, low time cost, and high precision. More importantly, it is not only deduced simply and directly, but also can be easily extended to the degree reduction of surfaces. Finally, we present two examples to demonstrate the effectiveness of our algorithm.

A novel algorithm for explicit optimal multi-degree reduction of triangular surfaces

This paper introduces the algebraic property of bivariate orthonormal Jacobi polynomials into geometric approximation. Based on the latest results on the transformation formulae between bivariate Bernstein polynomials and Jacobi polynomials, we naturally deduce a novel algorithm for multi-degree reduction of triangular B^zier surfaces. This algorithm possesses four characteristics： ability of error forecast, explicit expression, less time consumption, and best precision. That is, firstly, whether there exists a multi-degree reduced surface within a prescribed tolerance is judged beforehand; secondly, all the operations of multi-degree reduction are just to multiply the column vector generated by sorting the series of the control points of the original surface in lexicographic order by a matrix; thirdly, this matrix can be computed at one time and stored in an array before processing degree reduction; fourthly, the multi-degree reduced surface achieves an optimal approximation in the norm L2. Some numerical experiments are presented to validate the effectiveness of this algorithm, and to show that the algorithm is applicable to information processing of products in CAD system.

Numerical Study on Reduction in Aerodynamic Drag and Noise of High-Speed Pantograph

Reducing the aerodynamic drag and noise levels of high-speed pantographs is important for promoting environmentally friendly,energy efficient and rapid advances in train technology.Using computational fluid dynamics theory and the K-FWH acoustic equation,a numerical simulation is conducted to investigate the aerodynamic characteristics of high-speed pantographs.A component optimization method is proposed as a possible solution to the problemof aerodynamic drag and noise in high-speed pantographs.The results of the study indicate that the panhead,base and insulator are the main contributors to aerodynamic drag and noise in high-speed pantographs.Therefore,a gradual optimization process is implemented to improve the most significant components that cause aerodynamic drag and noise.By optimizing the cross-sectional shape of the strips and insulators,the drag and noise caused by airflow separation and vortex shedding can be reduced.The aerodynamic drag of insulator with circular cross section and strips with rectangular cross section is the largest.Ellipsifying insulators and optimizing the chamfer angle and height of the windward surface of the strips can improve the aerodynamic performance of the pantograph.In addition,the streamlined fairing attached to the base can eliminate the complex flow and shield the radiated noise.In contrast to the original pantograph design,the improved pantograph shows a 21.1%reduction in aerodynamic drag and a 1.65 dBA reduction in aerodynamic noise.

Single-atom catalysts for the electrochemical reduction of carbon dioxide into hydrocarbons and oxygenates

The electrochemical reduction of carbon dioxide offers a sound and economically viable technology for the electrification and decarbonization of the chemical and fuel industries.In this technology,an electrocatalytic material and renewable energy-generated electricity drive the conversion of carbon dioxide into high-value chemicals and carbon-neutral fuels.Over the past few years,single-atom catalysts have been intensively studied as they could provide near-unity atom utilization and unique catalytic performance.Single-atom catalysts have become one of the state-of-the-art catalyst materials for the electrochemical reduction of carbon dioxide into carbon monoxide.However,it remains a challenge for single-atom catalysts to facilitate the efficient conversion of carbon dioxide into products beyond carbon monoxide.In this review,we summarize and present important findings and critical insights from studies on the electrochemical carbon dioxide reduction reaction into hydrocarbons and oxygenates using single-atom catalysts.It is hoped that this review gives a thorough recapitulation and analysis of the science behind the catalysis of carbon dioxide into more reduced products through singleatom catalysts so that it can be a guide for future research and development on catalysts with industry-ready performance for the electrochemical reduction of carbon dioxide into high-value chemicals and carbon-neutral fuels.

Surface engineering of ZnO electrocatalyst by N doping towards electrochemical CO_(2) reduction

The discovery of efficient,selective,and stable electrocatalysts can be a key point to produce the largescale chemical fuels via electrochemical CO_(2) reduction(ECR).In this study,an earth-abundant and nontoxic ZnO-based electrocatalyst was developed for use in gas-diffusion electrodes(GDE),and the effect of nitrogen(N)doping on the ECR activity of ZnO electrocatalysts was investigated.Initially,a ZnO nanosheet was prepared via the hydrothermal method,and nitridation was performed at different times to control the N-doping content.With an increase in the N-doping content,the morphological properties of the nanosheet changed significantly,namely,the 2D nanosheets transformed into irregularly shaped nanoparticles.Furthermore,the ECR performance of Zn O electrocatalysts with different N-doping content was assessed in 1.0 M KHCO_(3) electrolyte using a gas-diffusion electrode-based ECR cell.While the ECR activity increased after a small amount of N doping,it decreased for higher N doping content.Among them,the N:ZnO-1 h electrocatalysts showed the best CO selectivity,with a faradaic efficiency(FE_(CO))of 92.7%at-0.73 V vs.reversible hydrogen electrode(RHE),which was greater than that of an undoped Zn O electrocatalyst(FE_(CO)of 63.4%at-0.78 V_(RHE)).Also,the N:ZnO-1 h electrocatalyst exhibited outstanding durability for 16 h,with a partial current density of-92.1 mA cm^(-2).This improvement of N:ZnO-1 h electrocatalyst can be explained by density functional theory calculations,demonstrating that this improvement of N:ZnO-1 h electrocatalyst comes from(ⅰ)the optimized active sites lowering the free energy barrier for the rate-determining step(RDS),and(ⅱ)the modification of electronic structure enhancing the electron transfer rate by N doping.

Model reduction of fractional impedance spectra for time–frequency analysis of batteries, fuel cells, and supercapacitors

Joint time–frequency analysis is an emerging method for interpreting the underlying physics in fuel cells,batteries,and supercapacitors.To increase the reliability of time–frequency analysis,a theoretical correlation between frequency-domain stationary analysis and time-domain transient analysis is urgently required.The present work formularizes a thorough model reduction of fractional impedance spectra for electrochemical energy devices involving not only the model reduction from fractional-order models to integer-order models and from high-to low-order RC circuits but also insight into the evolution of the characteristic time constants during the whole reduction process.The following work has been carried out:(i)the model-reduction theory is addressed for typical Warburg elements and RC circuits based on the continued fraction expansion theory and the response error minimization technique,respectively;(ii)the order effect on the model reduction of typical Warburg elements is quantitatively evaluated by time–frequency analysis;(iii)the results of time–frequency analysis are confirmed to be useful to determine the reduction order in terms of the kinetic information needed to be captured;and(iv)the results of time–frequency analysis are validated for the model reduction of fractional impedance spectra for lithium-ion batteries,supercapacitors,and solid oxide fuel cells.In turn,the numerical validation has demonstrated the powerful function of the joint time–frequency analysis.The thorough model reduction of fractional impedance spectra addressed in the present work not only clarifies the relationship between time-domain transient analysis and frequency-domain stationary analysis but also enhances the reliability of the joint time–frequency analysis for electrochemical energy devices.

Exploring the impact of Nafion modifier on electrocatalytic CO_(2) reduction over Cu catalyst

Nafion as a universal polymer ionomer was widely applied for nanocatalysts electrode preparation.However,the effect of Nafion on electrocatalytic performance was often overlooked,especially for CO_(2)electrolysis.Herein,the key roles of Nafion for CO_(2)RR were systematically studied on Cu nanoparticles(NPs)electrocatalyst.We found that Nafion modifier not only inhibit hydrogen evolution reaction(HER)by decreasing the accessibility of H_(2)O from electrolyte to Cu NPs,and increase the CO_(2)concentration at electrocatalyst interface for enhancing the CO_(2)mass transfer process,but also activate CO_(2)molecule by Lewis acid-base interaction between Nafion and CO_(2)to accelerate the formation of^(*)CO,which favor of C–C coupling for boosting C_(2)product generation.Owing to these features,the HER selectivity was suppressed from 40.6%to 16.8%on optimal Cu@Nafion electrode at-1.2 V versus reversible hydrogen electrode(RHE),and as high as 73.5%faradaic efficiencies(FEs)of C_(2)products were achieved at the same applied potential,which was 2.6 times higher than that on bare Cu electrode(~28.3%).In addition,Nafion also contributed to the long-term stability by hinder Cu NPs morphology reconstruction.Thus,this work provides insights into the impact of Nafion on electrocatalytic CO_(2)RR performance.

Preferentially selective extraction of lithium from spent LiCoO_(2)cathodes by medium-temperature carbon reduction roasting

Lithium recovery from spent lithium-ion batteries(LIBs)have attracted extensive attention due to the skyrocketing price of lithium.The medium-temperature carbon reduction roasting was proposed to preferential selective extraction of lithium from spent Li-CoO_(2)(LCO)cathodes to overcome the incomplete recovery and loss of lithium during the recycling process.The LCO layered structure was destroyed and lithium was completely converted into water-soluble Li2CO_(3)under a suitable temperature to control the reduced state of the cobalt oxide.The Co metal agglomerates generated during medium-temperature carbon reduction roasting were broken by wet grinding and ultrasonic crushing to release the entrained lithium.The results showed that 99.10%of the whole lithium could be recovered as Li2CO_(3)with a purity of 99.55%.This work provided a new perspective on the preferentially selective extraction of lithium from spent lithium batteries.

Optimum Profiles of Endwall Contouring for Enhanced Net Heat Flux Reduction and Aerodynamic Performance

Successfully utilized non-axisymmetric endwalls to enhance turbine efficiencies(aerodynamic and turbine inlet temperatures)by controlling the characteristics of the secondary flow in a blade passage.This is accomplished by steady-state numerical hydrodynamics and deep knowledge of the field of flow.Because of the interaction between mainstream and purge flow contributing supplementary losses in the stage,non-axisymmetric endwalls are highly susceptible to the inception of purge flow exit compared to the flat and any advantage rapidly vanishes.The conclusions reveal that the supreme endwall pattern could yield a lowering of the gross pressure loss at the design stage and is related to the size of the top-loss location being productively lowered.This has led to diminished global thermal exchange lowered in the passage of the vane alone.The reverse flow adjacent to the suction side corner of the endwall is migrated farther from the vane surface,as the deviated pressure spread on the endwall accelerates the flow and progresses the reverse flow core still downstream.The depleted association between the tornado-like vortex and the corner vortex adjacent to the suction side corner of the endwall is the dominant mechanism of control in the contoured end wall.In this publication,we show that the non-axisymmetric endwall contouring by selective numerical shape change method at most prominent locations is advantageous in lowering the thermal load in turbines to augment the net heat flux reduction as well as the aerodynamic performance using multi-objective optimization.

Comprehensive understanding of the thriving electrocatalytic nitrate/nitrite reduction to ammonia under ambient conditions

Ammonia(NH_(3))is a multifunctional compound that is an important feedstock for the agricultural and pharmaceutical industries and attractive energy storage medium.At present,NH_(3)synthesis is highly dependent on the conventional Haber–Bosch process that operates under harsh conditions,which consumes large quantities of fossil fuels and releases a large amount of carbon dioxide.As an alternative,electrosynthesis is a prospective method for producing NH_(3)under normal temperature and pressure conditions.Although electrocatalytic nitrogen reduction to ammonia has attracted considerable attentions,the low solubility of N_(2)and high N≡N cracking energy render the achievements of high NH_(3) yield rate and Faradaic efficiency difficult.Nitrate and nitrite(NO_(x)^(-))are common N-containing pollutants.Due to their high solubilities and low dissociation energy of N=O,NO_(x)^(-)−are ideal raw materials for NH_(3) production.Therefore,electrocatalytic NO_(x)^(-)−reduction to NH_(3)(eNO_(x)RR)is a prospective strategy to simultaneously realise environmental protection and NH_(3) synthesis.This review offers a comprehensive understanding of the thriving eNO_(x)RR under ambient conditions.At first,the popular theory and mechanism of eNO_(x)RR and a summary of the measurement system and evaluation criteria are introduced.Thereafter,various strategies for developing NO_(x)−reduction catalysts are systematically presented and discussed.Finally,the challenges and possible prospects of electrocatalytic NO_(x)^(-1) reduction are outlined to facilitate energy-saving and environmentally friendly large-scale synthesis of NH_(3) in the future.

Cu-Based Materials for Enhanced C_(2+) Product Selectivity in Photo-/Electro-Catalytic CO_(2) Reduction: Challenges and Prospects

Carbon dioxide conversion into valuable products using photocatalysis and electrocatalysis is an effective approach to mitigate global environmental issues and the energy shortages. Among the materials utilized for catalytic reduction of CO_(2), Cu-based materials are highly advantageous owing to their widespread availability, cost-effectiveness, and environmental sustainability. Furthermore, Cu-based materials demonstrate interesting abilities in the adsorption and activation of carbon dioxide, allowing the formation of C_(2+) compounds through C–C coupling process. Herein, the basic principles of photocatalytic CO_(2) reduction reactions(PCO_(2)RR) and electrocatalytic CO_(2) reduction reaction(ECO_(2)RR) and the pathways for the generation C_(2+) products are introduced. This review categorizes Cu-based materials into different groups including Cu metal, Cu oxides, Cu alloys, and Cu SACs, Cu heterojunctions based on their catalytic applications. The relationship between the Cu surfaces and their efficiency in both PCO_(2)RR and ECO_(2)RR is emphasized. Through a review of recent studies on PCO_(2)RR and ECO_(2)RR using Cu-based catalysts, the focus is on understanding the underlying reasons for the enhanced selectivity toward C_(2+) products. Finally, the opportunities and challenges associated with Cu-based materials in the CO_(2) catalytic reduction applications are presented, along with research directions that can guide for the design of highly active and selective Cu-based materials for CO_(2) reduction processes in the future.

Attribute Reduction Method Based on Sequential Three-Branch Decision Model

Attribute reduction is a research hotspot in rough set theory. Traditional heuristic attribute reduction methods add the most important attribute to the decision attribute set each time, resulting in multiple redundant attribute calculations, high time consumption, and low reduction efficiency. In this paper, based on the idea of sequential three-branch decision classification domain, attributes are treated as objects of three-branch division, and attributes are divided into core attributes, relatively necessary attributes, and unnecessary attributes using attribute importance and thresholds. Core attributes are added to the decision attribute set, unnecessary attributes are rejected from being added, and relatively necessary attributes are repeatedly divided until the reduction result is obtained. Experiments were conducted on 8 groups of UCI datasets, and the results show that, compared to traditional reduction methods, the method proposed in this paper can effectively reduce time consumption while ensuring classification performance.

A cascade of in situ conversion of bicarbonate to CO_(2) and CO_(2) electroreduction in a flow cell with a Ni-N-S catalyst

Combination of CO_(2) capture using inorganic alkali with subsequently electrochemical conversion of the resultant HCO_(3)^(-)to high-value chemicals is a promising route of low cost and high efficiency.The electrochemical reduction of HCO_(3)^(-)is challenging due to the inaccessible of negatively charged molecular groups to the electrode surface.Herein,we adopt a comprehensive strategy to tackle this challenge,i.e.,cascade of in situ chemical conversion of HCO_(3)^(-)to CO_(2) and CO_(2) electrochemical reduction in a flow cell.With a tailored Ni-N-S single atom catalyst(SACs),where sulfur(S)atoms located in the second shell of Ni center,the CO_(2)electroreduction(CO_(2)ER)to CO is boosted.The experimental results and density functional theory(DFT)calculations reveal that the introduction of S increases the p electron density of N atoms near Ni atom,thereby stabilizing^(*)H over N and boosting the first proton coupled electron transfer process of CO_(2)ER,i.e.,^(*)+e^(-)+^(*)H+^(*)CO_(2)→^(*)COOH.As a result,the obtained catalyst exhibits a high faradaic efficiency(FE_(CO)~98%)and a low overpotential of 425 mV for CO production as well as a superior turnover frequency(TOF)of 47397 h^(-1),outcompeting most of the reported Ni SACs.More importantly,an extremely high FECOof 90%is achieved at 50 mA cm^(-2)in the designed membrane electrode assembly(MEA)cascade electrolyzer fed with liquid bicarbonate.This work not only highlights the significant role of the second coordination on the first coordination shell of the central metal for CO_(2)ER,but also provides an alternative and feasible strategy to realize the electrochemical conversion of HCO_(3)^(-)to high-value chemicals.

Airfoil friction drag reduction based on grid-type and super-dense array plasma actuators

To improve the cruise flight performance of aircraft, two new configurations of plasma actuators(grid-type and super-dense array) were investigated to reduce the turbulent skin friction drag of a low-speed airfoil. The induced jet characteristics of the two actuators in quiescent air were diagnosed with high-speed particle image velocimetry(PIV), and their drag reduction efficiencies were examined under different operating conditions in a wind tunnel. The results showed that the grid-type plasma actuator was capable of producing a wall-normal jet array(peak magnitude: 1.07 m/s) similar to that generated in a micro-blowing technique, while the superdense array plasma actuator created a wavy wall-parallel jet(magnitude: 0.94 m/s) due to the discrete spanwise electrostatic forces. Under a comparable electrical power consumption level,the super-dense array plasma actuator array significantly outperformed the grid-type configuration,reducing the total airfoil friction drag by approximately 22% at a free-stream velocity of 20 m/s.The magnitude of drag reduction was proportional to the dimensionless jet velocity ratio(r), and a threshold r = 0.014 existed under which little impact on airfoil drag could be discerned.

Copper slag assisted coke reduction of phosphogypsum for sulphur dioxide preparation

The reduction of phosphogypsum(PG)to lime slag and SO_(2)using coke can effectively alleviate the environmental problems caused by PG.However,the PG decomposition temperature remains high and the product yield remains poor.By adding additives,the decomposition temperature can be further reduced and PG decomposition rate and product yield can be improved.However,the use of current additives such as Fe_(2)O_(3)and SiO_(2)brings the problem of increasing economic cost.Therefore,it is proposed to use solid waste copper slag(CS)as a new additive to reduce PG to prepare SO2,which can reduce the cost and meet the environmental benefits at the same time.The effects of proportion,temperature and thermostatic time on PG decomposition are investigated by experimental and kinetic analysis combined with FactSage thermodynamic calculations to optimize the roasting conditions.Finally,the reaction mechanism is proposed.It is found that adding CS to the coke and PG system can increase the rate of PG decomposition and SO_(2)yield while lowering the PG decomposition temperature.For example,when the CS/PG mass ratio increases from 0 to 1,PG decomposition rate increases from 83.38%to 99.35%,SO_(2)yield increases from 78.62%to 96.81%,and PG decomposition temperature decreases from 992.4℃to 949.6℃.The optimal reaction parameters are CS/PG mass ratio of 1,Coke/PG mass ratio of 0.06 at 1100℃for 20 min with 99.35%PG decomposition rate and 96.81%SO_(2) yield.The process proceeds according to the following reactions:2CaSO_(4)+ 0.7C + 0.8Fe_(2)SiO_(4)→0.8Ca_(2)SiO_(4)+ 0.2Ca_(2)Fe_(2)O_(5)+ 0.4Fe_(3)O_(4)+2SO_(2)+ 0.7CO_(2)Finally,a process for decomposing PG with coke and CS is proposed.