Lagrange Interpolation Method Research Articles

A Study on the Modelling of the State Variables for Superheated and Saturated Region

The state variables of saturated and superheated region in the steam table were simultaneously modeled using the neural networks. And the results were compared with quadratic spline interpolation and Lagrange interpolation. Two input data without distinguishing parameter were used in the neural networks. For comparison, quadratic spline interpolation method for superheated region and Lagrange interpolation method for saturated region were applied. The overall results revealed that the neural networks were greatly superior to quadratic interpolation method or Lagrange interpolation method.

Hybrid [formula omitted] with Lagrange multiplier and upwinding for the multiscale transport capability in Rattlesnake

Two interface conditions, the Lagrange multiplier method and the upwinding method, for hybrid PN-SN calculations are proposed for the self-adjoint angular flux (SAAF) formulation of the transport equation using the continuous finite element method (FEM) for spatial discretization. These interface conditions are implemented in Rattlesnake, the radiation transport application built on MOOSE, for the on-going multiscale transport simulation effort at INL. For smoothing the solution at the interface for the Lagrange multiplier method, a SN Lagrange interpolation method on the unit sphere is proposed. The discrete PN interface condition with upwinding method for coupling the SN solution is desired. Numerical results indicate that the interface conditions give the expected convergence.

Application of interleaved circuitry to improve evaporator effectiveness and COP of a packaged AC system

Highly constrained air flow pathways as experienced in tightly packaged air conditioning systems result in air flow maldistribution problems in the evaporators. The interleaved circuitry method, where the refrigerant from a circuit with high air flow is routed to a circuit with low air flow and vice-versa, has been investigated to passively reduce the air maldistribution effect. Air velocity measurements have been conducted in psychrometric chambers and the measurement locations have been defined by the log-Tchebycheff rule. The velocity profile was obtained by Lagrange Interpolation method as percentage values. The performance of the interleaved circuitry method was compared to the baseline circuitry at different operating conditions. The results show that the interleaved circuitry method uniforms the superheat of the individual circuits and improves the cooling capacity and COP by up to 16.6% and 12.4%, respectively. Furthermore, the tuned model predicted the evaporator cooling capacity within a mean absolute error of approximately ±10%.

REDUCED SIMULATION OF THE WIND FIELD BASED ON HERMITE INTERPOLATION

Owing to the problem that the computational efficiency of traditional spectral representation (SR) method by Deodatis is restricted to the times of Cholesky decompositions, Hermite interpolation is included in the decomposition of cross-spectral density matrix. A reduced method for wind field simulation is derived accordingly. With the reduced method, the time of Cholesky decomposition in the traditional SR method can be decreased from n×N to n×2k (2k<N), hence the efficiency will be sharply increased. Taking the wind field simulation of the girder of a long-span triple-tower suspension bridge as an example, a 4096s-long fluctuating wind field is simulated by traditional SR method, third-order Lagrange interpolation method and the derived Hermite interpolation method. The comparison of the computational efficiency and its corresponding accuracy indicate that both the third-order Lagrange interpolation method and the Hermite interpolation method can prominently improve the computational efficiency of SR method. The efficiency of the Hermite interpolation method is slightly lower than that of the third-order Lagrange one, but its computing accuracy of Matrix H is on a much higher level which shows great superiority in wind field simulation. When the reduced method with Hermite interpolation is adopted, both the power spectral density and correlation functions of the simulated wind field can well conform to the targets. Based on the work above, the suggested boundaries for the interpolation intervals are given according to an optimization analysis.

Derivation of Adams Method for the Numerical Solution of Ordinary Differential Equations Via Computer Algebra

In this paper, a well-known computer algebra system (CAS) was considered for the derivation of Numerical method for the solution of initial value problems. This was achieved by the use of maple software. Numerical methods were derived through Lagrange interpolation method. Both the implicit and explicit method was derived with the help of the Computer algebra system. In particular, a review of Maple’s functional role in the derivation of numerical methods was also presented. The main challenge was that the efficient handling and simplifying of very long expressions, which was met by the power of Maple’s build-in functionality. The use of the maple procedure had significantly reduced the errors and hence improved efficiency in derivation of higher order Adams Methods.

The relationship of degenerate kernel and projection methods on Fredholm integral equations of the second kind

In this paper, we show that the degenerate kernel method for some cases, on the condition that the source function is approximated by the same way of producing degenerate kernel, becomes as a projection method. We consider two ways, including Lagrange interpolation and best approximation methods, of producing degenerate kernel approximations of more general Fredholm integral equation of the second kind. For these two ways, we show that the degenerate kernel method becomes as a Lagrange-collocation method and Galerkin method respectively.

Comparison of Selected Methods of Probe Radius Correction Based on Measurements of Ceramic Workpieces

The article concerns the analysis of results of coordinate measurements of ceramic products. The measured parts were made of ZrO2 based ceramic material and machined using the CNC machine tool Ultrasonic 20 linear. The analysis of results of measurements of ceramic workpieces concerned the comparison of corrected measured points obtained by means of different algorithms of the probe radius correction process. The first algorithm was developed based on the Lagrange interpolation method of indicated measured points. The second method was the commercial one and it is available in the Calypso software which can cooperate with a coordinate measuring machine.

A Novel Multi-Receiver Signcryption Scheme with Complete Anonymity.

Anonymity, which is more and more important to multi-receiver schemes, has been taken into consideration by many researchers recently. To protect the receiver anonymity, in 2010, the first multi-receiver scheme based on the Lagrange interpolating polynomial was proposed. To ensure the sender’s anonymity, the concept of the ring signature was proposed in 2005, but afterwards, this scheme was proven to has some weakness and at the same time, a completely anonymous multi-receiver signcryption scheme is proposed. In this completely anonymous scheme, the sender anonymity is achieved by improving the ring signature, and the receiver anonymity is achieved by also using the Lagrange interpolating polynomial. Unfortunately, the Lagrange interpolation method was proven a failure to protect the anonymity of receivers, because each authorized receiver could judge whether anyone else is authorized or not. Therefore, the completely anonymous multi-receiver signcryption mentioned above can only protect the sender anonymity. In this paper, we propose a new completely anonymous multi-receiver signcryption scheme with a new polynomial technology used to replace the Lagrange interpolating polynomial, which can mix the identity information of receivers to save it as a ciphertext element and prevent the authorized receivers from verifying others. With the receiver anonymity, the proposed scheme also owns the anonymity of the sender at the same time. Meanwhile, the decryption fairness and public verification are also provided.

Weight Functions and Stress Intensity Factors for Eccentric through Cracks in a 3-D Rectangular Plate Subjected to In-Plane Loading

Three unknown coefficient weight functions for eccentric through cracks in a 3-D rectangular plate subjected to in-plane loading are proposed. 3-D finite element models of cracked rectangular plates within the whole range of crack aspect ratios, i.e., 0≤e/W≤0.8, 0.08≤a/(W-e)≤0.9, were established to obtain a reference SIF database for both crack points A and B, rather than 2-D finite element models. To improve the accuracy of the weight function, the coefficients were derived from this database using the Binary Lagrange Interpolation Method instead of Curve-Fitting Expression. Comparisons of stress intensity factors calculated using the present weight functions with finite element data for the high-order power law and residual stress distributions show high accuracy of the present weight functions.

English

Burning the fuel required to sustain a given flight releases pollution such as carbon dioxide and nitrogen oxides, and the amount of fuel consumed is also a significant expense for airlines. It is desirable to reduce fuel consumption to reduce both pollution and flight costs. To increase fuel savings in a given flight, one option is to compute the most economical vertical reference trajectory (or flight plan). A deterministic algorithm was developed using a numerical aircraft performance model to determine the most economical vertical flight profile considering take-off weight, flight distance, step climb and weather conditions. This algorithm is based on linear interpolations of the performance model using the Lagrange interpolation method. The algorithm downloads the latest available forecast from Environment Canada according to the departure date and flight coordinates, and calculates the optimal trajectory taking into account the effects of wind and temperature. Techniques to avoid unnecessary calculations are implemented to reduce the computation time. The costs of the reference trajectories proposed by the algorithm are compared with the costs of the reference trajectories proposed by a commercial flight management system using the fuel consumption estimated by the FlightSim® simulator made by Presagis®.

Lagrange Interpolation Learning Particle Swarm Optimization.

In recent years, comprehensive learning particle swarm optimization (CLPSO) has attracted the attention of many scholars for using in solving multimodal problems, as it is excellent in preserving the particles’ diversity and thus preventing premature convergence. However, CLPSO exhibits low solution accuracy. Aiming to address this issue, we proposed a novel algorithm called LILPSO. First, this algorithm introduced a Lagrange interpolation method to perform a local search for the global best point (gbest). Second, to gain a better exemplar, one gbest, another two particle’s historical best points (pbest) are chosen to perform Lagrange interpolation, then to gain a new exemplar, which replaces the CLPSO’s comparison method. The numerical experiments conducted on various functions demonstrate the superiority of this algorithm, and the two methods are proven to be efficient for accelerating the convergence without leading the particle to premature convergence.

Vehicle sideslip estimation and compensation for banked road

The sideslip driving status is of fundamental importance to the stability of a vehicle. This paper presents a practical vehicle sideslip driving status estimation method that uses ESP (electronic stability program) sensors. ESP sensors such as wheel speed, lateral acceleration, yaw rate and steering wheel angle sensors are used to determine the sideslip driving status and distinguish a banked road. This estimation algorithm contains front-rear sideslip and banked road detection methods. The proposed sideslip estimation algorithm was designed to use the analytical redundancy of these sensors and Lagrange interpolation methods. The performance and effectiveness of the proposed estimation and compensation algorithm were investigated using vehicle tests. This paper presents the results of two cases that were used for the experimental verification: a curved flat road and banked road.

The prediction of the elastic critical load of submerged eccentric cylindrical shell based on vibro-acoustic model

An effective new approach to nondestructively predict the elastic critical hydrostatic pressure of a submerged eccentric cylindrical shell is presented in this paper. According to the geometry characteristic of the cross section of eccentric cylindrical shell, the eccentric problem is transformed into the problem of variable thickness in the circumferential direction. The vibration equations considering hydrostatic pressures of outer fluid are written in the form of a matrix differential equation which is obtained by using the transfer matrix of the state vector of the shell. Depending on wave propagation approach, the data of the fundamental natural frequencies of the shell with various eccentricities under different hydrostatic pressure and boundary conditions are obtained by solving the frequency equation with a Lagrange interpolation method. The curve of the fundamental natural frequency squared versus hydrostatic pressure is then drawn with the data, which is approximately straight line or parabola that depends on the eccentric value. The elastic critical hydrostatic pressure is therefore obtained while the fundamental natural frequency is assumed to be zero according to the curve. The results obtained by the present approach show good agreement with published results and finite element results.

Lagrange Interpolation for Upsampling

In this paper, we compare well known interpolation methods such as nearest neighbor, bilinear, bicubic, triangle kernel, and Lagrangian interpolation method. Reconstruction errors from above interpolation methods are compared using test image. From the simulation results, it can be found that Lagrangian method outperforms all other upsampling methods. Visual performance comparison is provided by using two LC images.

Modelling surface tension with smoothed particle hydrodynamics in reactive rotational moulding

Reactive Rotational Moulding (RRM) is the best process for producing large hollow plastic parts without weld lines. Constant quality in technical parts requires the process to be mastered by controlling on-line the main physical phenomena. However, the main drawback of RRM is poor control of the process due to the high number of influent parameters. In these conditions, the optimization of the process is quite complex. The aim of this work is to simulate the reactive fluid flow during RRM with Smoothed Particle Hydrodynamics (SPH) solver in two dimensions (2D) and three dimensions (3D) taking into account surface tension force. To implement this force, the interface is tracked explicitly using algorithm developed by Barecasco et al. (2013) and Terissa et al. (2013) and the reconstruction of curve or surface boundary by different interpolation or surface construction technique with Lagrangian interpolation and fitting circle methods in 2D and spherical regression in 3D, respectively.

Prediction of the Aging of HTV Silicone Rubber Using Chemical Concentration and Polynomial Interpolation Approach

Predicting the lifetime of polymeric insulators is one of the most important research topics in studying the life cycle of high voltage insulators in the power transmission and distribution networks. HTV (high temperature vulcanized) silicone rubber is a high performance dielectric material used within electrical power equipment, in particular transmission and distribution insulators. In this paper, we proposed a new approach using the Newton's method and Lagrange method to predict the aging of HTV silicone rubber that are subjected to multiple stress conditions. Concentration of chemical elements such as carbon, oxygen, silicon and aluminum were obtained and evaluated using a SEM (scanning electron microscope) with EDS (energy dispersive X-ray spectroscopy). Curve fitting using the Newton's and Lagrange interpolation methods yield useful linear interpolation equations that describe the aging characteristic of the specimens under study. This approach can be applied to predict the change in chemical concentration of polymeric insulators over the life cycle with good accuracy.

The quality control method for remanufacturing assembly based on the Jacobian-torsor model

To improve the assembly accuracy of remanufactured parts with multiple heterogeneity and properties, and to guarantee the quality of remanufacturing product no worse than the original one, this paper takes the air tightness of remanufactured engines’ cylinder block and head as the research target and proposes a quality control method based on the Jacobian-torsor model for remanufacturing assembly. First of all, three different Jacobian-torsor models for assembly tolerance are structured, and according to that, the defined torsor difference (∆FR) of remanufacturing products is calculated by considering the geometric error of remanufactured related parts assembly influenced by the temperature field and force field. Then, the optimal tightening torque of cylinder heads’ main bolts corresponding to ∆FR is achieved by applying the bivariate Lagrange interpolation method. Finally, taking the assembly of remanufactured engines’ cylinder block and head as a case study to verify the proposed methods is feasible and effective.

Trajectory planning for energy minimization of industry robotic manipulators using the Lagrange interpolation method

We propose to use Lagrange interpolation method to express each joint trajectory function to realize trajectory planning for energy minimization of industrial robotic manipulators. We give the position constraints to the industrial robotic manipulators and the stability of the industrial robotic manipulators should be satisfied. In order to avoid Runge’s phenomenon of Lagrange interpolation method, we use the Chebyshev interpolation points for our approach. Through derivation, the angular velocity functions and angular acceleration functions can be obtained. Lagrange interpolation method can satisfy position constraints, and ensure the smoothness of joint angular positions, velocities, accelerations, and joint torques. By taking these functions into the Performance Index (PI) of energy minimization, as well as the direct iteration method used for optimization of energy consumption, we can obtain the optimal trajectory for industrial robotic manipulators.

Anonymous multi‐receiver ID‐based signcryption scheme

In earlier multi-receiver signcryption schemes, in order to help the authorised receivers to find the corresponding information for decryption from the ciphertext, the list of identities of all authorised receivers should be included in the ciphertext as a necessary element. However, this method has directly led to the privacy leakage problem of receivers. Later, several anonymous multi-receiver signcryption schemes use the Lagrange interpolation method to mix and hide the real identities of all receivers, so that the ciphertext does not expose identities of the receivers any longer. Therefore the Lagrange interpolation method is considered to be a perfect solution to the receiver anonymity problem, and multiple anonymous multi-receiver signcryption schemes have been proposed based on it. However, recently, this method has been proven to be insecure because it cannot assure the anonymity of receivers either. In fact, in any of the existing anonymous schemes based on the Lagrange interpolation method, it is easy for one authorised receiver to judge whether anyone is an authorised receiver or not, which badly violates the receiver anonymity. Motivated by these concerns, the authors propose a new multi-receiver signcryption scheme to solve this problem. Analyses show that the proposed scheme meets confidentiality, unforgeablity and the sender and receivers’ anonymity.

라그랑지 보간법을 이용한 로봇 매니퓰레이터의 토크 최소화를 위한 궤적계획

Abstract This paper proposes an algorithm using Lagrange interpolation method to realize trajectory planning for torque minimization of robot manipulators. For the algorithm, position constraints of robot manipulators should be given and the stability of robot manipulators should be satisfied. In order to avoid Runge’s phenomenon, we set up time interpolation points using Chebyshev interpolation points. After that, we found suitable angle which correspondsto the points and then we got trajectories of joint's angle, velocity, acceleration using Lagrange interpolation method. We selected performance index for torque consumption optimization of robot manipulator. The method went through repetitive computation process to have minimum value of the performance index by calculated trajectory. Throughthe process, we could get optimized trajectory to minimize torque and performance index and guarantee safety of the motion for manipulator performance. Key Words : Iterative method, Lagrange interpolation method, Performance index, Trajectory planning, Torque minimization