Degree Of Bending Research Articles

Bio-Inspired Hydrogel-Elastomer Actuator with Bidirectional Bending and Dynamic Structural Color.

In nature, some creatures can change their body shapes and surface colors simultaneously to respond to the external environments, which greatly inspired researchers in the development of color-tunable soft actuators. In this work, we present a facile method to prepare a smart hydrogel actuator that can bend bidirectionally and change color simultaneously, just like an octopus. The actuator is fabricated by elastomer/hydrogel bilayer and the hydrogel layer was decorated with thermoresponsive microgels as the photonic crystal blocks. Compared with the previously reported poly(N-isopropylacrylamide) hydrogel-based bilayer hydrogel actuators, which are generally limited to one-directional deformation, the elastomer/hydrogel bilayer actuator prepared in our work exhibits unique bidirectional bending behavior in accordance with the change of structural color. The bending degrees can be changed from -360° to 270° in response to solution temperatures ranging from 20 °C to 60 °C. At the same time, the surface color changes from red to green, and then to blue, covering the full visible light spectrum. The bending direction and degree of the hydrogel actuator can easily be adjusted by tuning the layer thickness ratio of the elastomer/hydrogel or the composition of the hydrogel. The color-tunable hydrogel-elastomer actuator reported in this work can achieve both programmable deformations and color-changing highly resembling the natural actuating behaviors of creatures.

Study of flexible metasurface with gold rod resonators on ploymide substrate in terahertz waveband

A flexible metasurface based on gold rod resonator arrays on a polyimide substrate in the terahertz band is presented. Four different resonator structures with various azimuth angle were designed and investigated. Experimental results demonstrate that the azimuth angle of the gold rod resonators have a significant impact on the terahertz waves under different polarization conditions. Experimental studies on the bending performance reveal that the spectral modulation characteristics of the flexible metasurface is robust within a degree of bending. In addition, the effect of the incident angle on the resonant peaks was investigated, and the redshift of the high-frequency resonant peak was found as the incident angle increases.

A reactive transport model for in-situ leaching of weathered crust elution-deposited rare earth ores using ammonium sulfate

This study presents a reactive transport model for the in-situ recovery of weathered crust elution-deposited rare earth ores and aims to provide predictive and optimization techniques for rare earth recovery. The model integrates Kerr's selection coefficient, dynamic reversible shrinking core model, and transport model to capture the dynamics of leaching fraction, rare earth elements (REEs), and exchangeable ions during in-situ leaching. The proposed model was validated through bench-scale equilibrium, kinetic, and column experiments, as well as numerical simulation. The results indicate that the deviation from using a single selection coefficient (referred to as 4.26 L2 mol−2 in this study) for modeling is acceptable. Both Darcy velocity and concentration of leaching solution mainly affect the peak value and bending degree of the post-peak leaching curve. Optimal cessation of leaching solution injection, such as substituting water after the peak of the leaching curve, enables effective conservation of leaching solution and reduction of environmental pollution. Differences of at least two orders of magnitude were observed between dynamic parameters derived from kinetic experiments (1.46 and 4.34 L mol−1 min−1) and those obtained from column experiments (2.60 × 10−3 and 1.71 × 10−2 L mol−1 min−1). This finding carries substantial implications for in-situ recovery and mineralization processes, highlighting the need for further investigation at the pore scale.

Construction of amorphous V2O5@Ti3C2Tx synergistic heterostructure on 3D carbon cloth substrate by a self-assembled strategy towards high-performance aqueous Zn-ion batteries

The emerging aqueous Zn-ion batteries (ZIBs) with their low price and inherent safety are showing strong competitiveness in the energy storage field. In this work, layer-by-layer stacked amorphous V2O5@Ti3C2Tx heterostructures anchored on three-dimensional carbon cloth (3D CC), synthesized by annealing process and subsequent in-situ electrochemical induction, are reported as high capacity and stable cathode for ZIBs. In this composite cathode, amorphous V2O5@Ti3C2Tx synergistic heterostructure provides isotropic Zn2+ migration channels and rapid electron transfer kinetics, while the 3D CC substrate can ensure that the bulk phase in the electrode material is also utilized to maximize the synergistic effect. As a result, the CC/a-V2O5@Ti3C2Tix cathode achieves a high specific capacity of 567 mAh/g at 0.1 A/g, a satisfactory rate performance of 213 mAh/g at 10 A/g, together with a long cycle life of 96.2 % capacity retention after 2000 cycles. Ex-situ characterizations show that the CC/a-V2O5@Ti3C2Tx cathode undergoes a highly reversible Zn2+/H+ co-intercalation/deintercalation mechanism during the discharge/charging process. In addition, based on the CC/a-V2O5@Ti3C2Tx cathode, the assembled flexible ZIB can operate properly under different bending degrees, showing its practicality in flexible devices. This work provides insight into the rational design of cathode materials with high Zn-storage performance.

Efficient Photocatalytic Reduction of Hexavalent Chromium by BiVO4-Decorated MXene Photocatalysts and Their Charge Carrier Dynamics.

The synergistically MXene (Ti3C2Tx) co-catalyst-decorated BiVO4-based heterostructured photocatalysts have been synthesized by a hydrothermal approach with varied loading concentrations of MXene (Ti3C2Tx) to drive the hexavalent chromium reduction efficiently. The formation of the heterostructured photocatalyst was confirmed by the appearance of X-ray diffraction (XRD) peaks corresponding to the monoclinic BiVO4 phase and MXene (Ti3C2Tx) and also the antisymmetric (834 cm-1) and symmetric stretching (715 cm-1) of tetrahedral VO4 and D (1330 cm-1) and G (1570 cm-1) bands corresponding to MXene (Ti3C2Tx) in the Raman spectrum. The worm-like structures of BiVO4 nanocrystals grew onto the lamellar sheets of MXene (Ti3C2Tx), as shown by field emission scanning electron microscopy (FESEM), and has an increased surface area of 15.62 m2g-1 in the case of BVO-20-TC. X-ray photoelectron spectroscopy (XPS) analysis confirms the presence of V5+ and Ti3+states, and the uniform distribution of BiVO4 nanocrystals over lamellar sheets of MXene (Ti3C2Tx) is evident from energy-dispersive X-ray (EDX) analysis. The ultraviolet-diffuse reflectance spectroscopy (UV-DRS) spectra suggest a decrease in the band gap energy of BVO-20-TC to 2.335 eV, promoting a higher degree of visible light harvesting. Upon optimization, by varying the pH, the amount of the photocatalyst, and the concentration of Cr(IV), BVO-20-TC exhibits the highest photocatalytic efficiency (96.39%) while using a Cr(VI) concentration of 10 ppm at pH 2 and 15 mg of the photocatalyst, and the photoreduction of Cr(VI) to Cr(III) follows the pseudo-first-order reaction. The decrease in the PL intensity in BVO-20-TC reveals a faster transfer of electrons from MXene (Ti3C2Tx) to BiVO4. Further, the higher degree of band bending at the BiVO4/MXene (Ti3C2Tx) heterojunction, revealed from the Mott-Schottky analysis, facilitates efficient charge transfer and eventually faster and efficient photoreduction of Cr(VI) to Cr(III). The reusability and stability test undertaken for BVO-20-TC reveals that even after five cycles, the Cr (VI) photoreduction efficacy is retained. This work provides insights into photoreduction of Cr (VI) by using such heterostructures.

Scale Effect of a Kappel Tip-Rake Propeller

In this paper, the scale effect of Kappel tip-rake propellers with different end plate designs was studied using computational fluid dynamics. Given the base size of the mesh and the appropriate numerical model for the determined simulation, the open-water performance of three Kappel propellers with different bending degrees of the end plate at different scales was calculated. Comparing the scale effect of these propellers, the scale effect of the torque coefficient of a Kappel propeller is more intense than that of the conventional propeller. In addition, the scale effect of the torque coefficient is strong when the bending degree of the end plate increases, dwarfing the scale effect on the thrust coefficient. Following the research on the scale effect of the wake field for the Kappel propeller, the laws that reveal the influence of the scale on the wake field were summarized; that is, the high-speed zone in the wake relatively expands with the increase of the scale in company with a trend of tip cross flow. The research reveals the basic variation trend and rule of the open-water performance and wake distribution for the Kappel propeller under different scales within the Reynolds number range of 4.665×105−8.666×107 considering γ transition, as well as the characteristic differences between the Kappel propellers with different end plate designs, which will be of great significance to its optimization design and application to marine vehicles of different scales.

Inhomogeneity of Cleaved Bulk MoS2 and Compensation of Its Charge Imbalances by Room‐Temperature Hydrogen Treatment

AbstractSynthetic single crystals of bulk molybdenum disulphide cleaved in ultrahigh vacuum are mapped across a large (≈25 mm2) area by X‐ray photoelectron spectroscopy, both statically and transiently following above‐bandgap excitation by an ultrafast laser. This work finds that: I) A cleaved surface typically displays spatially inhomogeneous properties, manifested by large (≈1 eV) variations in binding energy and band bending and variable degrees of stability of those over time as a result of variable gas uptakes from the residual atmosphere. II) Moderate (350 °C) annealing and exposure to molecular hydrogen can be cycled to switch between smaller and larger surface band bending, the switch being reversible but strongly sample‐position dependent. III) Upon exposure to atomic hydrogen, the binding energy of the entire surface levels out to a common (within <0.05 eV) value corresponding to a Fermi level pinned close to mid‐bandgap. Such remarkable effect is attributed to the ability of hydrogen atoms to serve as donors and acceptors alike, thus neutralizing local charge imbalances inevitably present at the surface in consequence of intrinsic and/or cleavage‐induced defects. With subsequent moderate annealing, the hydrogenated surface preserves a fairly homogenous electronic state which is however characterized by a lower binding energy and little to no band bending.

Crystal Structures of DNA Intercalating Agents Dipyrido[3,2-f:2′,3′-h]quinoxaline (dpq), (Benzo[i]dipyrido[3,2-a:2′,3′c]phenazine (dppn), and [Ir(ppy)2(dppn)][PF6] (Where Hppy = 2-Phenylpyridine)

Pyrazino-phenanthroline ligands are commonly used with transition metals as DNA intercalation agents. Herein, we report the characterization of two commonly utilized pyrazino-phenanthroline ligands, dipyrido[3,2-f:2′,3′-h]quinoxaline (dpq) and (benzo[i]dipyrido[3,2-a:2′,3′c]phenazine (dppn), by single-crystal X-ray diffraction. Additionally, the characterization of [Ir(ppy)2(dppn)][PF6], where Hppy = 2-phenylpyridine, by single-crystal X-ray diffraction is described. Both the dpq and dppn ligands crystallize as chloroform solvates where the chloroform molecule occupies the equivalent binding pocket of a metal in metal complexes of these ligands. These pyrazino-phenanthrolines are largely planar, with the dppn ligand displaying a slight twist. When the dppn ligand is coordinated to iridium(III), the dppn ligand on the resulting complex displays a significant degree of bending along the longitudinal direction of the ligand. This iridium (III) complex crystallizes as a CH2Cl2 and Et2O solvate and due to the volatility of these solvents these crystals are only stable for a few seconds outside of the mother liquor. The structures of the free ligands and the [Ir(ppy)2(dppn)][PF6] complex all display extensive π stacking interactions.

(Invited) highly Stretchable Self-Powered Sensors Enabled By Vertically-Assembled 2D Materials

Future electronics, such as human health monitoring systems, require highly stretchable strain sensors applicable to various types and ranges of motions. Two-dimensional (2D) materials exhibiting superior mechanical properties have a potential to achieve high range of strain compared to conventional devices and also realize self-powered sensing. In this talk, we present self-powered strain sensor with large stretchability enabled by graphene electrode layers and molybdenum disulfide (MoS2) active layer. In order to maximize the energy harvesting ability, flexoelectric effect was adopted in our device design. Flexoelectric effect induces an electric polarization from strain gradient, and the size of the strain gradient grows quadratically as the material’s length scale decreases. This scaling enables maximum flexoelectric effects in 2D materials with strain gradient. To realize the strain gradient in MoS2 active layer, we first prepared graphene-MoS2 vertical stack on a pre-stretched elastomeric substrate followed by a release of the stretch to create a crumpled heterostructure. The crumpled MoS2/graphene was again stretched to a smaller strain, and the other graphene electrode layer was placed on top of the crumpled MoS2 active layer. Releasing the strain after assembly resulted in a periodic contact between the top graphene electrode and the MoS2 active layer while maintaining a full contact between the bottom graphene electrode and the MoS2 active layer. Our stretchable flexoelectric graphene/MoS2/graphene device demonstrated voltage and current outputs under various mechanical deformations, including bending and stretch-release motions. Our results demonstrate self-powered strain sensing capability under high degree of bending and stretching motions, potentially applicable to human health monitoring sensors.

Triphenylamine-Based Helical Polymer for Flexible Memristors.

Flexible nonvolatile memristors have potential applications in wearable devices. In this work, a helical polymer, poly (N, N-diphenylanline isocyanide) (PPIC), was synthesized as the active layer, and flexible electronic devices with an Al/PPIC/ITO architecture were prepared on a polyethylene terephthalate (PET) substrate. The device showed typical nonvolatile rewritable memristor characteristics. The high-molecular-weight helical structure stabilized the active layer under different bending degrees, bending times, and number of bending cycles. The memristor was further employed to simulate the information transmission capability of neural fibers, providing new perspectives for the development of flexible wearable memristors and biomimetic neural synapses. This demonstration highlights the promising possibilities for the advancement of artificial intelligence skin and intelligent flexible robots in the future.

Negative pressure-assisted porous structure with gradient dielectrics design for linearity enhancement of flexible capacitance pressure sensor

The trade-off between sensitivity and linearity range for flexible capacitive pressure sensors remains unresolved. Herein, a simple and low-cost method is proposed to fabricate porous structure with the aid of negative pressure. The gradient dielectrics are assembled with porous structure doped with different CNT ratios. With the perforated design on the low dielectric layer, the gradient dielectric can realize series-parallel conversion during the pressure-induced deformation process, where the coverage ratio and dielectric properties of low dielectric layer can be appropriately collocated to realize the linear behavior of the capacitance performance over the full-scale pressure. Therefore, the capacitive sensor based on optimized gradient dielectric layer can achieve a high sensitivity of 0.247 kPa−1 and a wide linearity range of 0–175 kPa. With the high sensitivity over a broad pressure range, the capacitive sensor demonstrates its various practical applications in monitoring physiological signals and human motions such as the artery pulse, respiratory, knee bending, elbow bending degree and walking and running status, showing potential in wearable healthcare monitor.

THE INFLUENCE OF THE USUAL HOLDING SOLUTIONS ON Hyacinthus orientalis CUT FLOWER VASE LIFE

In this study, four holding solutions very often used by florists and the final consumer were tested to analyse their impact on the development of the inflorescence, as well as the elongation, weight and degree of bending of the hyacinth flowers. The use of these solutions has a general character, and the flower species react differently due to both their genetics and the conditions in which they were cultivated, handled and stored. To verify the effectiveness of the preservative solutions on Hyacinthus orientalis vase life, four solutions were prepared with 2% sucrose and none or one of the follow-ing antimicrobial substances: sodium hypochlorite, acetic acid and ascorbic acid. During the study, measurements were made on the weight, elongation of the floral stems, chlorophyll content of the leaves and vase life days. The vase life of the hyacinths in this experiment was 5 days, except for the flowers from the 2% sucrose holding solution, which started to wilt on day 5. On day 6, the flowers showed signs of senescence. The increase in the length of the inflorescence stem showed significant differences for the sucrose + ascorbic acid holding solution, with a maximum of 0.8 cm on day 4. However, the interest was not to maximise the elongation of the floral stem but to slow down this process since elongation leads to the bending of the stem, causing it to require additional support.

Centrifugal and gyroscopic effects on dynamic response of rotating cantilever beams under step loading

In this paper, a simplex finite element model for beams (FEMB) with a complete formulation is presented to study the dynamics of rotating cantilever beams subjected to distributed external loads. In our approach, the finite element method (FEM) and Timoshenko's beam theory is used. We will, particularly, examine the centrifugal and gyroscopic effects on the linear vibration of a rotating cantilever beam in stationary regime. In this model, extension, bending, and torsion degrees of freedom (DOF) are combined using a parameterization of the 3D motion of the beam by Euler's three-angle sequence. That allows identifying all the terms of the gyroscopic coupling in a more compact equations system. To ensure convergence, a sufficient number of finite elements are required in this model. The considered beam in the numerical simulation is pre-twisted and linearly tapered with a rectangular section. Results show that the dynamic centrifugal effect decreases the natural frequencies of the beam. Furthermore, the gyroscopic coupling induces rapid extension and torsion beatings when the beam undergoes a step bending load. Without damping, these beatings can persist and cause material fatigue.

Analysis of Adolescents’ Head to Shoulder Region during Tablet Use from Sagittal and Frontal RGB Images

As schools go digital, the use of tablet computers is increasing. Concerns are raised that the extensive use of tablets and the associated bent-over posture may negatively affect the individual’s health. In order to analyse the possible effects of prolonged tablet use on physical health, a detailed analysis of the posture during tablet use is needed so that appropriate preventive measures can be taken to prevent degenerative changes. Therefore, the aim of this study was to measure and report the posture of 56 students while working with a tablet computer and compare it with an upright posture. Sagittal and frontal images were used for measurements of the subjects’ postures while seated, using the tablet, and in a neutral sitting position looking straight ahead. The body position during tablet use was recorded in two different user configurations: tablet flat on the table and tablet in individual freely chosen user configuration. After appropriate annotation of the data, the following parameters were evaluated in different planes. The craniovertebral angle (CVA), head tilt angle (HTA), and forward shoulder angle (FSA) are measurements that describe the extent to which the head bends forward and downward and how the shoulders are aligned in the sagittal plane. On the other hand, the head shoulder angle (HSA), lateral head tilt angle (LHTA), and trunk flexion angle (TFA) are angles measured in the frontal plane, which indicate the degree of head tilt and trunk bending to the right or left side. The measurement results clearly showed that the use of a tablet had a pronounced effect on the positions and rotations of the participants’ head, neck, and shoulders. This was evident through strong deviations observed in the angles measured between the sitting straight posture and the postures while using the tablet. For example, depending on the body posture class, the mean CVA values were 45.76° for straight sitting posture, 28.25° for holding the tablet individually posture, and 26.04° for the posture adopted while using a tablet placed flat on the table.

Improving flexural toughness of foamed concrete by mixing polyvinyl alcohol-polypropylene fibers: An experimental study

In this study, the bending properties of polyvinyl alcohol-polypropylene fiber toughened foamed concrete were assessed by the three-point bending test. The flexural properties of specimens were studied in terms of load–deflection curve, failure mode, toughness, and toughening mechanism. The results showed that the mixture of various toughening materials had a synergistic effect on improving the bending property of polyvinyl alcohol-polypropylene fiber toughened foamed concrete. In addition, the increase of the contents of polyvinyl alcohol and polypropylene fibers could lead to the improvement of ductility, strength degradation, bending toughness, and bending degree of foamed concrete after reaching the peak. The influence of polypropylene fiber content on the specimen was greater than that of polyvinyl alcohol content. The toughened specimen had a single crack fracture mode. According to the scanning electron microscopy observations, the polyvinyl alcohol-polypropylene fiber mixture could provide an integrated network with satisfactory cracking resistance and form a variety of energy dissipation modes, thereby improving the bending toughness of the specimen.

Effect of Scratching and Bending Degree on the Residual Stress of the 7055 Aluminum Alloy Plate

This study investigated the effects of scratches and bending on the surface stress of aluminum alloy plates with varying thicknesses. The purpose of this study was to provide guidance for surface stress testing of plates with different thicknesses. The “birth and death” element method was used to simulate the generation of surface scratches, while the stress field under the pre-stretched state was obtained through the displacement of one end of the board by 2.5%. The X-ray residual stress analyzer (μ-x360 s) was used to experimentally characterize the stress level on the surface of the plate. The research results showed that the residual stress level near the scratches decreases significantly, while the surface stress level of the plate decreases significantly after bending. The changes in stress values with increasing bending degree for plates of different thicknesses showed that the thicker the plate, the greater the tensile effect on the surface of the plate, resulting in smaller compressive stress. The longitudinal non-uniformity of the plates with different thicknesses was also analyzed, and it was found that the testing error of stress testing results was within ±5 MPa. In conclusion, this study provides useful guidance for the accurate measurement of surface stress on aluminum alloy plates of varying thicknesses, considering the effects of scratches, and bending.

Design of Intelligent Robots Based on Body Sense Control

Recognizing human actions to control the robot has become a challenging task. This paper provides a manipulator control system based on a mobile platform. The bending Angle of the specific connecting rod mechanism is collected to obtain the bending degree of the finger. The nine-axis gyroscope is used to detect the attitudes of the hand and the arm, and the data is transmitted to the HT32F5352 control board. Experimental results show that intelligent robots based on motion sensing can be equipped with automatic and manual control, environmental detection, and other functions.

Using Highly Flexible SbSn@NC Nanofibers as Binderless Anodes for Sodium-Ion Batteries

Flexible and binderless electrodes have become a promising candidate for the next generation of flexible power storage devices. However, developing high-performance electrode materials with high energy density and a long cycle life remains a serious challenge for sodium-ion batteries (SIBs). The main issue is the large volume change in electrode materials during the cycling processes, leading to rapid capacity decay for SIBs. In this study, flexible electrodes for a SnSb alloy–carbon nanofiber (SnSb@NC) membrane were successfully synthesized with the aid of hydrothermal, electrospinning and annealing processes. The as-prepared binderless SnSb@NC flexible anodes were investigated for the storage properties of SIBs at 500 °C, 600 °C and 700 °C (SnSb@NC-500, SnSb@NC-600 and SnSb@NC-700), respectively. And the flexible SnSb@NC-700 electrode displayed the preferable SIB performances, achieving 240 mAh/g after 100 cycles at 0.1 A g−1. In degree-dependent I-V curve measurements, the SnSb@NC-700 membrane exhibited almost the same current at different bending degrees of 0°, 45°, 90°, 120° and 175°, indicating the outstanding mechanical properties of the flexible binderless electrodes.

Design of Stable 2D Heterostructured Photocatalyst for Drastically Improved Hydrogen Generation and Unravelling the Underlying Charge Carrier Dynamics

AbstractThe design of heterostructured photocatalysts comprising of 2‐D mesoporous C3N3 (π‐conjugated bonds with one less nitrogen atom in the ring to promote the better electron transport unlike conventional g‐C3N4) donor coupled with the MoS2 acceptor by a template‐directed thermal synthesis followed by hydrothermal synthesis is presented. The formation and coupling of MoS2 through basal planes on the surface of mpg‐CN has been confirmed by the XRD, XPS (C/N ratio of the order of 1), and lengthening of the C−N bond as revealed by increasing intensity of FTIR peak (1232 cm−1), respectively. Upon optimization, MS‐mpg‐CN exhibits 10‐fold enhancement in hydrogen generation rate and efficiency (∼86500 μmolg−1 h−1, IPCE: ∼64 %) than that of observed for MS‐mpg‐C3N4 photocatalyst (∼8250 μmolg−1 h−1, IPCE: ∼5.6 %). The lower PL emission intensity, average lifetime (1.91 ns) of charge carriers, precise semicircle in Nyquist plot and higher degree of band bending (Efb −0.64 eV), higher ‘g’ value (2.0038) in the EPR imply improved charge carrier separation, transfer kinetics and formation of intimate heterojunction, respectively. Furthermore, hydrogen evolution measured for subsequent days while reusing photocatalyst shows no significant change in the activity (∼1.2 % variation over three consecutive cycles), confirming their photostability and reusability.

The nonlinear effects of spinning on the dynamics of a pitching cantilever

We investigate the structural dynamics of a pitching cantilever as it spins about a fixed axis, emulating the motion of the rotor blades of wind turbines or rotorcraft to some extent. From first principles, we derive a system of equations consisting of a nonlinear partial differential equation (PDE) coupled with a nonlinear ordinary differential equation (ODE) that describes the motion of the system in bending and pitch as it spins according to a prescribed rotor velocity and acceleration. The equations of motion consider the interaction between dynamics of a flexible body in bending, rigid body motion in pitch, and the nonlinear effects of spinning, concurrently. This simplified model of a rotor blade permits an intuitive understanding of the various nonlinear terms that arise in the equations of motion, and their relative influence on the system dynamics. In a series of numerical experiments, we study the effects of the coupling between the bending and pitch degrees of freedom, the influence of rotor velocity, and rotor acceleration on the system dynamics. We compare the full nonlinear simulation results to those of the underlying linear system, obtained by imposing a small angle assumption in pitch and excluding all remaining nonlinear terms. The numerical experiments are performed by non-dimensionalizing the equations, discretizing the non-dimensionalized equations in space by applying the Galerkin method, discretizing the resulting system of ODEs in time using the Houbolt method and finally solving the resulting system of implicit algebraic equations using Newton’s method.