Cosine Law Research Articles

Estimation of the radiation field for CPC photocatalytic reactors using a novel six-flux model in two dimensions (SFM-2D)

In the present work, we introduced a new Six Flux Model in two dimensions (SFM-2D) for the estimation of the local volumetric rate of photon absorption (LVRPA) in polar coordinates. This novel model was applied on a compound parabolic collector (CPC) photocatalytic reactor using the Ray-Tracing Technique (RTT) and Lambert's cosine law. The volumetric rate of photon absorption per unit reactor length (VRPA/H) and overall volumetric rate of photon absorption (OVRPA), calculated with SFM-2D, were compared with SFM-1-DR (diffuse reflectance) and SFM-HG (Henyey-Greenstein).We obtained an analytical solution of the radiant transfer equation (RTE) based on a novel SFM-2D. The most relevant result was a simple algebraic expression from a detailed mathematical development. In addition, we included Lambert's cosine law and the radiation absorption by the fluid to estimate the radiant field in a compound parabolic photoreactor CPC. The numerical results, obtained with simulations, were compared satisfactorily to those reported in previous works at different operating conditions; the global performance of the new model described discrepancies less than 0.5%.

Effect of curvature correction on parameters extracted from hyperspectral images.

.Significance: Hyperspectral imaging (HSI) has emerged as a promising optical technique. Besides optical properties of a sample, other sample physical properties also affect the recorded images. They are significantly affected by the sample curvature and sample surface to camera distance. A correction method to reduce the artifacts is necessary to reliably extract sample properties.Aim: Our aim is to correct hyperspectral images using the three-dimensional (3D) surface data and assess how the correction affects the extracted sample properties.Approach: We propose the combination of HSI and 3D profilometry to correct the images using the Lambert cosine law. The feasibility of the correction method is presented first on hemispherical tissue phantoms and next on human hands before, during, and after the vascular occlusion test (VOT).Results: Seven different phantoms with known optical properties were created and imaged with a hyperspectral system. The correction method worked up to 60 deg inclination angle, whereas for uncorrected images the maximum angles were 20 deg. Imaging hands before, during, and after VOT shows good agreement between the expected and extracted skin physiological parameters.Conclusions: The correction method was successfully applied on the images of tissue phantoms of known optical properties and geometry and VOT. The proposed method could be applied to any reflectance optical imaging technique and should be used whenever the sample parameters need to be extracted from a curved surface sample.

Approximate universality in the electric field variation on a field-emitter tip in the presence of space charge

The electric field at the surface of a curved emitter is necessary to calculate the field emission current. For smooth parabolic emitting tips where space charge is negligible, variation of the electric field at the surface is known to follow the generalized cosine law. Here, we investigate the validity of the cosine law in the regime, where space charge due to emitted electrons is important. Particle-in-cell (PIC) simulations with an emission algorithm based on the cosine law are employed for this study. It is shown that if EP and EL be the field at the apex of tip with and without space charge, respectively, then for ϑ=EP/EL≥0.9, the average relative deviation of the electric field from the cosine law is less than 3% over the endcap. Thus, an emission scheme based on the cosine law may be used in PIC simulations of field emission of electrons from curved emitter tips in the weak space charge regime. The relation between ϑ and normalized current ζ for curved emitters in this regime is also investigated. A linear relation, ϑ=1−δζ (where δ is a constant), similar to that obtained theoretically for flat emitting surfaces is observed, but the value of δ indicates that the extension of the theory for curved emitters may require incorporation of the field enhancement factor.

Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths

We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space [Formula: see text]. It can be determined by the set of six edge lengths up to isometry. For further considerations, we use the notion of edge matrix of the tetrahedron formed by hyperbolic cosines of its edge lengths. We establish necessary and sufficient conditions for the existence of a tetrahedron in [Formula: see text]. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formula expressing the volume of a hyperbolic tetrahedron in terms of the edge lengths. The latter volume formula can be regarded as a new version of classical Sforza’s formula for the volume of a tetrahedron but in terms of the edge matrix instead of the Gram matrix.

Crosswind effects on a train-bridge system: wind tunnel tests with a moving vehicle

To better understand the aerodynamic characteristics of a train-bridge system under crosswind, an in-house experimental apparatus was developed in a wind tunnel at Central South University (CSU). The new facility consists of trains, a bridge, U-shaped slide rails, and positioning systems with winches. A wireless data acquisition system for pressure measurements was mounted within the train carriage to avoid any potential aerodynamic interference. A 1:25 scaled model of CRH2 running over a simply supported girder bridge was tested in a wind tunnel. The aerodynamic characteristics of the train-bridge system under various combinations of crosswind speed, train speed, and vehicle location on the bridge were identified. The experimental results reveal that the cosine law for surface wind pressure on a train under crosswind is no longer valid when the train runs on the bridge. The moving model method can better capture the more practical aerodynamic characteristics and the impact of train movement on the bridge. The present study indicates that the wind tunnel experiments using the stationary model cannot fully reveal the aerodynamic characteristics of both the train and bridge during crossing. The results captured using this unique approach provide insight into the aerodynamic characteristics of trains and bridges.

Localization in Wireless Sensor Networks with Mobile Anchor Node Path Planning Mechanism

Determining the accurate locations of randomly deployed sensors in a given geographical region is a critical problem in the field of Wireless Sensor Networks. Various approaches have been proposed in the past to tackle this problem. Most of the existing techniques rely on the use of a beacon node which traverses throughout the region with limited power sources. Sensors use the beacon node positions for self localization. The trajectory of the beacon node has an influence on the accuracy, time and efficiency of the localization algorithm. In this paper, we propose Cosine Rule based Localization (CRL) algorithm for static trajectories of the mobile beacon node. Existing schemes use trilateration for position estimation of sensor nodes. The proposed method uses the Cosine rule on received beacon positions and distances obtained from RSSI in such a way that lines representing distances intersect at one point for the application of the Cosine rule. Simulation results reveal that CRL enables the unknown sensors to locate themselves with better accuracy in comparison with the existing trilateration technique for all the trajectories.

Propagation factor of partially coherent radially polarized vortex beams in anisotropic turbulent atmosphere.

We skillfully combined the cosine theorem with the second moment theory and the Wigner distribution function and derived the analytical expressions of the propagation factor (M2-factor) of a partially coherent radially polarized vortex beam (PCRPVB) in atmospheric turbulence. Then, we comparatively studied the propagation factors of a PCRPVB and a partially coherent electromagnetic vortex beam (PCEVB) in atmospheric turbulence. The results show that a PCRPVB has a smaller value of a relative M2-factor than a PCEVB, which means that a PCRPVB has a stronger ability to resist atmospheric turbulence than a PCEVB. To confirm our theoretical studies, the hyperbolic fitting method is combined with the random phase screen (RPS) to simulate the M2-factor of a PCRPVB and a PCEVB through atmospheric turbulence. The study results indicate that the theoretical values agree well with the simulated values. Our results may find applications in free-space optical communications and remote sensing.

Nanostructure optimization of Zr-W-Ti metallic glass thin films via multitarget co-sputtering with oblique angle deposition approach

In this work, we systematically investigated the Zr-W-Ti nanostructured metallic glasses (NMGs) thin films on silicon (Si) wafer substrate fabricated via multitarget co-sputtering with oblique angle deposition (OAD) technique. By varying the substrate angles (θs) from 0° to 55°, the evolution of Zr-W-Ti NMGs from dense to nanocolumnar and nanorod film were observed by field-emission scanning electron microscope (FE-FEM), which created by the self-shadowing effect. The critical point of substrate angles was around 15° to produce the nanocolumnar and 40° to form the nanorod structure. The relationship among the substrate angles and the Zr-W-Ti NMGs tilted angle (β) was determined and discussed in detail with two theoretical correlations as the "Tangent rule" and "Cosine rule." The X-ray photoelectron spectroscopy (XPS) studied revealed the different chemical compositions. The results obtained from grazing incidence X-ray diffraction (GIXRD) and transmission electron microscope (TEM) indicated that the prepared Zr-W-Ti NMGs thin films exhibit an amorphous structure. Ultimately, it suggests that the multitarget co-sputtering with the OAD technique provides NMGs thin films, enabling applications in gas storage and sensor devices due to their well-defined nanostructured morphological and chemical composition characteristics.

Development of a geometric slip flat model when turning a vehicle with two steering axles

Introduction (problem statement and relevance). One of the main stages in the design ofspecial purpose vehicles is the calculation of the steering control. At that, engineers are guided by anumber of regulatory documents that lack one of the most important requirements, which is to minimize tire lateral deviation. The author notes the lack of scientific research in the field of geometric slip, which is caused by the non-compliance between the actual angles of wheels rotation and the calculated values for pure rolling and is an inherent property of any traditional steering linkage.The purpose of the study was to develop a mathematical model of the steering drive of a special-purpose vehicle with two steering axles to assess the geometric and power slip.Methodology and research methods. There is a known method for calculating the steering drive using trigonometric expressions, in particular the cosine theorem. The author proposed to use the coordinateiterative method developed by him and based on the equation of the sphere, with the steering wheel rotation angle in the kinematic calculation of the steering drive as a differentiation step. The choice of the steering drive parameters according to the conditions of symmetry and minimization of slip was carried out by the method of multivariable optimization.Results. In the course of the research, it was found that the choice of the characteristic of geometric noncompliance was a multi-parameter task, and changing one parameter led to the necessity of adjusting the others. If it was not possible to achieve zero geometric slip for all steered wheels, the task of optimizing the steering drive parameters wasreduced to minimizing geometric or total slip. The value of the slip essentially depended on the selected differentiation step. When choosing the characteristic of geometric slip, it was necessary to observe the condition of the steering linkage symmetry when turning left and right. When the wheels were turned from the neutral position to the periphery, the power and geometric slip compensated each other, which led to the decrease of the total slip and tire wear.The scientific novelty of the work lies in the development of a geometric slip model for a vehicle with two steerable axles, including a spatial model of the steering drive which allows to evaluate the influence of the geometric slip on the turn kinematics, as well as the mutual influence of geometric and power slip in order to select the steering drive optimal parameters of the multi-axle vehicle from the viewpoint of minimizing tire wear during curvilinear motion.Practical significance. The research results must be taken into account in the development of steering drive and turning control systems for multi-axle special-purpose vehicles, including them in the educational process as well.

Fusion of Multi-Sensor-Based Biomechanical Gait Analysis Using Vision and Wearable Sensor

The purpose of this research work is to investigate the biomechanics of pelvis, hip, knee, and ankle joint motion using a Kinect sensor & inertial measurement unit (IMU) sensor during the normal walk. In this paper, a very cost-effective gait analysis system based on Microsoft Kinect v2 and Inertial Measurement Unit (IMU) device is presented. Kinect sensor is used for acquiring 3D skeleton data (camera (x, y, z), depth (x, y) orientation (x, y, z, w), color (x, y)) with 25 human body joints. For this analysis, the lower extremities joints i.e. spinal cord joint, hip, knee, and ankle joints of both left and right legs are being considered. The main contribution of this research work is the joint angle calculation of lower extremities of human gait based on Microsoft Kinect sensor V2 and IMU sensor. From the law of cosine, the joint angle is calculated between the two joints and plotted for a single subject. We came with the observation that the characteristics of the human knee joint and ankle joint are inversely related to each other. There are two sharp humps for knee and ankle joints during the normal walk. During the swing phase, the knee joint is highly activated while during toe-off and heel strike it is least activated. This analysis of clinical data is very useful for prosthesis limb and exoskeleton design. The stability of calculated joints trajectories is validated using the limit cycle curve. A system is designed for real-time analysis of biomechanics of different lower limbs joints using gait.

Uncertainty Estimation in Hydrogeological Forecasting with Neural Networks: Impact of Spatial Distribution of Rainfalls and Random Initialization of the Model

Neural networks are used to forecast hydrogeological risks, such as droughts and floods. However, uncertainties generated by these models are difficult to assess, possibly leading to a low use of these solutions by water managers. These uncertainties are the result of three sources: input data, model architecture and parameters and their initialization. The aim of the study is, first, to calibrate a model to predict Champagne chalk groundwater level at Vailly (Grand-Est, France), and, second, to estimate related uncertainties, linked both to the spatial distribution of rainfalls and to the parameter initialization. The parameter uncertainties are assessed following a previous methodology, using nine mixed probability density functions (pdf), thus creating models of correctness. Spatial distribution of rainfall uncertainty is generated by swapping three rainfall inputs and then observing dispersion of 27 model outputs. This uncertainty is incorporated into models of correctness. We show that, in this case study, an ensemble model of 40 different initializations is sufficient to estimate parameter uncertainty while preserving quality. Logistic, Gumbel and Raised Cosine laws fit the distribution of increasing and decreasing groundwater levels well, which then allows the establishment of models of correctness. These models of correctness provide a confidence interval associated with the forecasts, with an arbitrary degree of confidence chosen by the user. These methodologies have proved to have significant advantages: the rigorous design of the neural network model has allowed the realisation of models able to generalize outside of the range of the data used for training. Furthermore, it is possible to flexibly choose the confidence index according to the hydrological configuration (e.g., recession or rising water table).

Angular dependence of vertical force and torque when magnetic dipole moves vertically above flat high-temperature superconductor* *Projects supported by the National Natural Science Foundation of China (Grant No. 11572232) and the China Three Gorges Corporation Research Project (Grant No.202103407).

The interaction between a permanent magnet (PM) assumed as a magnetic dipole and a flat high-temperature superconductor (HTS) is calculated by the advanced frozen-image model. When the dipole vertically moves above the semi-infinite HTS, the general analytical expression of vertical force and that of torque are obtained for an arbitrary angle between the magnetization direction of the PM and the c axis of the HTS. The variations of the force and torque are analyzed for angle and vertical movements in both zero-field cooling (ZFC) condition and field cooling (FC) condition. It is found that the maximum vertical repulsive or attractive force has the positive or negative cosine relation with the angle. However, the maximum torque has the positive or negative sine relation. From the viewpoint of the rotational equilibrium, the orientation of the magnetic dipole with zero angle is deemed to be an unstable equilibrium point in ZFC, but the same orientation is considered as a stable equilibrium point in FC. In addition, both of the variation cycles of the maximum force and torque with the angle are π.

Geographic Information Systems Practice Midwife In Merauke Regency Android-Based Method Using Spherical Law of Cosines

Health is a resource for the daily life of the community. Because without health, a person can not run the activity with the good. To that end, the society can not be separated from the figure of health that play a role in manifesting the life of a healthy society as a midwife. One of the reasons people really need a midwife is the cost of the examination or the cost of labor is affordable. Therefore, it is needed a Geographic Information System Midwife Practice in Merauke Regency Android-Based Method Using Spherical Law of Cosines can help and facilitate user to search, and know the location of the midwife practice and service information available at the place of such practices as well as the closest distance to the user. Geographic Information system Midwife Practice in Merauke Regency Android Based on the design by using Eclipse with the java programming language, and using a MySQL database. The method used to search the nearest distance of the user with the location of the midwife practice using the method of Spherical Law of Cosines, and the method of testing using the method of blackbox and questionnaires. The result of this research is to produce a geographic information system application based on android to handle mapping the location of the midwife practice as well as data information midwife online which is accessible to the public in Merauke Regency.

A Novel Algorithm for Leaf Incidence Angle Effect Correction of Hyperspectral LiDAR

As a novel remote sensor, hyperspectral LiDAR is faced with the incidence angle effect, which restricts its quantitative applications. However, the current radiometric correction algorithms have some limitations, concentrating on: 1) the mathematically polynomial fitting; 2) adjacent wavelength ratio such as ratio vegetation index; and 3) perfect Lambertian assumption and using the Lambert cosine law to correct the effect. In this study, a practical and proper correction algorithm is proposed to overcome these limitations. First, to better characterize the complex reflection characteristics of the object surface, a combination of the Lambert law and Beckmann law is applied to represent the object surface. Then, it considers the impact of both wavelength and incidence angle on describing the surface roughness factor and diffuse fraction. Finally, a modified and physically based radiometric correction algorithm is generated. It provides the detailed correction equations for intensity and reflectance data recorded by hyperspectral LiDAR. To obtain its parameters and verify its reliability, leaves (ten samples per species, 30 in total) were stochastically collected from three broadleaf trees to experiment. The results showed that the algorithm achieved good performances by comparing the intensity and reflectance changes before and after removing the leaf incidence angle effect. Since it is physically based, the algorithm could promisingly be a fundamental solution to eliminate the incidence angle effect for hyperspectral LiDAR.

Experimental investigation of the effects of length-to-diameter ratio of hot-wire sensor on the dynamic response to velocity fluctuations

This paper reconsidered the criterion on the length-to-diameter ratio (l/d) of the hot-wire sensor, $$l/d \ge 200$$ , for end-conduction effects on the dynamic response of constant-temperature hot-wire anemometer (CTA) to be neglected. Experiments were first conducted using a low-noise compact CTA with hot-wire sensors of l/d = 80, 120, 150 and 200 in a two-dimensional (2-D) circular cylinder wake at a Reynolds number Re = 77, at which a 2-D von Karman vortex street developed, allowing us to exclude spatial filtering and tangential velocity effects. The damping coefficient in the CTA bridge was tuned to 0.70 for all sensors to reduce possible influences of over- or under-damping on measurements in a frequency range up to 6 kHz. The stubs and prongs of probes consisted of identical materials and dimensions so that the thermal properties were considered to be identical for all probes. Careful comparisons of the amplitudes of the fundamental (1.9 kHz) and higher harmonic components of time-periodic wake vortices showed that the attenuation due to the end-conduction effect in the absence of the spatial filtering and damping coefficient effects was at most 2.5% in terms of the rms value of velocity fluctuations even for l/d = 80. The similar weak attenuation (about 2.6% for l/d = 80) was also observed in the measurements of low-frequency free-stream turbulence. In addition, effects of l/d on the yaw angle response of the sensor were examined in the free stream, revealing that the yaw angle ( $$\psi $$ ) variation of the measured velocity (both the mean and fluctuations) obeyed the cosine law for $$\psi \le 60^\circ $$ . The Champagne constant k was 0.12 for l/d = 200 and 0.2 even for l/d = 80 in the mean velocity measurement using the present probe with long stubs. On the other hand, the intensity of free stream turbulence measured using a $$45^\circ $$ -yawed sensor was significantly affected by the sensor’s l/d; the rms value of velocity fluctuation was attenuated by about 7.5% for l/d = 80.

The Similarity of Essay Examination Results using Preprocessing Text Mining with Cosine Similarity and Nazief-Adriani Algorithms

Exams are one way to measure the level of students' ability to participate in learning. One type of exam given to students is the essay type. This study focuses on making automatic assessments for essay-type exams using cosine similarity. This method has several stages such as folding Case, tokenizing, filtering, stemming, analyzing, weighing of words in documents with cosine similarity. The stemming process uses the Nazief & Adriani algorithm. The results of this study are to conclude that the choice of words that are considered as keywords in the answer key greatly affects the results of the system's assessment. This is evidenced by testing applying the cosine law of 89.5%. However, there are several types of questions that are significantly different because there are unique characters in the database and answer keys that do not contain keywords that match the correct answer.

Memorization Techniques for Peter Chew Rule

We normally use sine rule to find the opposite side angle given when we are given two angles and one side. We also can use sine rules to find non-included angle when we are given two sides and a non-included angle. However, sine rule is not suitable to find the included angle when we are given three sides or to find the third side when we are given two sides and the included angle. This is because if we use the sine rule, the equation will have 2 unknowns. As we know, an equation with 2 unknowns cannot be solved. Therefore, we need to learn Cosine rule to solve the above problem. When giving 2 sides and an included angle, the sine rule and the cosine rule also are not suitable for finding the non-included angle directly. This is because if you use a sine rule or a cosine rule, the equation will have 2 unknowns. As mentioned earlier, equations with 2 unknowns cannot be solved. Therefore, we need to learn Peter Chew rule to solve the above problem. Peter Chew Rule allows us to find non included angles directly, more easier and more accurate. Symmetry, we can get 3 similar sine rules, 3 similar cosine rules and 6 similar Peter Chew rules. A Memorization Techniques must be created to make us happily remember 6 similar Peter Chew rules. For sine rule, a/sin⁡A = b/sin⁡B = c/sin⁡C . Techniques to remember rules is sides versus angles. If the angle we are looking for is A, then the side is also a. As same, If the angle we are looking for is B, then the side is also b. For cosine rule, a^2 = b^2 〖+ c〗^2 -2 bc cos⁡A. If the side we are looking for is a ,then the angle given normally is also A .As with the sine and cosine rules, the memorization techniques to remember Peter Chew rules, A = (a sin B)/(c - a cos B) , is also sides versus angles. If the angle we are looking for is A, then for the right hand side, the side is also a. If the angle is known as B, then for the denominator, the other side is c. the denominator will be c - a cos B. If the angle is known as C, the other side of the denominator is b, the denominator will be b - a cos C. Therefore, Peter Chew rule is tan A = (a sin C)/(b - a cos C) . So the technique to remember is very simple, the denominator will involve a, b, c. Therefore, remembering Peter Chew's rules is as easy as remembering a, b, and c. A game will be developed to test the efficiency of Peter Chew Rule's memorization techniques.

Peter Chew Triangle Diagram and Application

The objective of Peter Chew Triangle Diagram is to clearly illustrate the topic solution of triangle and provide a complete design for the knowledge of AI age. Peter Chew's triangle diagram will suggest a better single rule that allows us to solve any problem of topic solution of triangle problems directly, more easily and more accurately. There are two important rules for solving the topic solution of triangle today [1,2], namely the sine rule and the cosine rule. The sine rule is used to find a non-included angle when are given two sides and a non-included angle or the opposite side angle given when are given two angles and one side. The cosine rule normally is used to find the included angle when are given three sides or the third side when are given two sides and the included angle. Generally, we only think that when given two sides and an included angle, the cosine rule is used to find the third side. In fact, when two sides and one non included angle are given, the cosine rule is also more easier for finding the third side. For problem given 2 sides and an included angle, directly find the non included angle. We need to use Peter Chew rule [1] to solve this problem. Peter Chew Rule allows us to find the non included angles directly, easier and more accurately. The application of Peter Chew's triangle diagram in the PCET calculator allows the PCET calculator to directly solve any problem in the topic solution of triangle, which is easier and more accurate. The Peter Chew diagram provides a complete design of the topic solution of triangle, which can help students solve any problems in the topic solution of triangle directly, more easily, and more accurately. Apply Peter Chew diagram to the new calculator (PCET calculator) , allows the PCET calculator to solve any problems in the topic solution of triangle and solve some problem that can not solve by current online calculator such as Math Portal and Symbolab. Which can make PCET calculator effectively help the teaching of mathematics, especially when similar covid-19 problems arise in the future.

Polya's Problem Solving Strategy in Trigonometry: An Analysis of Students' Difficulties in Problem Solving

This study is focused on investigating errors made by students and the various causal factors in working on trigonometry problems by applying sine and cosine rules. Samples were taken randomly from high school students. Data were collected in two ways, namely a written test that was referred to Polya's strategy and interviews with students who made mistakes. Students' errors were analyzed with the Newman concept. The results show that all types of errors occurred with a distribution of 3.83, 19.15, 24.74, 24.89 and 27.39% for reading errors (RE), comprehension error (CE), transformation errors (TE), process skill errors (PSE), and encoding errors (EE), respectively. The RE, CE, TE, PSE, and EE are marked by errors in reading symbols or important information, misunderstanding information and not understanding what is known and questioned, cannot change problems into mathematical models and also incorrectly use signs in arithmetic operations, student inaccuracies in the process of answering and also their lack of understanding in fraction operations, and the inability to deduce answers, respectively. An anomaly occurs because it turns out students who have medium trigonometry achievements make more mistakes than students who have low achievement.

Regional iterative phase extraction Dixon water-lipid separation method based on second order differential quality weighting

To propose a regional iterative phase extraction Dixon method based on second order difference quality weighting (SOD-RIPE) for improving water-lipid separation in heterogeneous magnetic field. The in-phase angle of the asymmetric in-phase and opposite phase image matrix was eliminated using the Dixon's signal model to obtain J1 and J2, from which the possible water signal magnitude (B, S) and in-phase and opposite phase difference (P1, P2) was derived using the Cosine law. The phase quality map R of J2 was calculated using the second-order difference, and the phase difference was weighted to obtain (Pv, Pu). The Pv and Pu were iteratively selected to obtain the Pf. The water-fat separation diagram (W, F) was obtained using the least square method to bring the Pf into the Dixon signal model. Water-lipid separation was performed using 1000 pairs of in- and opposite-phase images on a Philips in Genia II 3.0T magnetic resonance scanner. The SOD-RIPE algorithm achieved better separation and stability than the automatic growth method and RIPE in all the parts of the body and in the stability test, and had a similar performance to mDixon-XD algorithm. SOD-RIPE can achieve robust water-fat separation with a good stability and can be used as a general Dixon water-fat separation method.