Numerical Integration Process Research Articles

Yarkovsky Effect on the Orbital Dynamics of 1566 Icarus Asteroid

The orbital dynamic of small objects is an n-body problem that can not be solve by analitically, it is needed to use numerical integration to find the solution instead. This work is about orbital dynamic of asteroid 1566 Icarus under Classical Newtonian gravitation and if thermal effect (Yarkovsky effect) is included. Yarkovsky Effect is a thermal radiation force resulted from time span of small rotating objects to receive heat from the Sun and then re-radiates it. The Yarkovsky Effect is working optimum for objects with diameter from 10 cm up to 10 km, and now is implemented to Asteroid 1566 Icarus with diameter 1.3 km which are member of Apollo and Earth crosser object. This Asteroid is called Earth crosser due to its orbit is crossing Earth’s orbit. With semi major axis a  1.078 au and eccentricity e  0.827, asteroid 1566 Icarus has perihelion distance q = 0.18674 au or less than semi major axis of Mercury. Due to that reason, Yarkovsky effect was considered to be applied on the orbital dynamics of asteroid 1566 Icarus. Due to sensitivity in input-data of numerical integration for n-body, one hundred simulation preliminary data were made as input in numerical integration process, therefore, 100 clones of Asteroid 1566 Icarus are gathered. Cloning process was conducted by using random number from Asteroid 1566 Icarus orbital elements at epoch 2456800.5 (23 May 2014) to standard deviation . The integration then later conducted within 105 years time span from the epoch. The result shown that the orbital dynamics of asteroid 1566 Icarus with Yarkovsky effect is still within the range of 100 clones of asteroid 1566 Icarus. Thereby, within 105 years, Yarkovsky effect does not change the orbital dynamic of asteroid 1566 Icarus globally, except for two phenomenon between close encounter with planet.

A Transient Stability Numerical Integration Algorithm for Variable Step Sizes Based on Virtual Input

In order to reduce the online calculations for power system simulations of transient stability, and dramatically improve numerical integration efficiency, a transient stability numerical integration algorithm for variable step sizes based on virtual input is proposed. The method for fully constructing the nonhomogeneous virtual input for a certain integration scheme is given, and the calculation method for the local truncation error of the power angle for the corresponding integration scheme is derived. A step size control strategy based on the predictor corrector variable step size method is proposed, which performs an adaptive control of the step size in the numerical integration process. The proposed algorithm was applied to both the IEEE39 system and a regional power system (5075 nodes, 496 generators) in China, and demonstrated a high level of accuracy and efficiency in practical simulations compared to the conventional numerical integration algorithm.

A state space approach for the eigenvalue problem of marine risers

A numerical state-space approach is proposed to examine the natural frequencies and critical buckling limits of marine risers. A large axial tension in the riser model causes numerical limitations. These limitations are overcome by using the modified Gram–Schmidt orthonormalization process as an intermediate step during the numerical integration process with the fourth-order Runge–Kutta scheme. The obtained results are validated against those obtained with other numerical methods, such as the finite-element, Galerkin, and power-series methods, and are found to be in good agreement. The state-space approach is shown to be computationally more efficient than the other methods. Also, we investigate the effect of a high applied tension, a high apparent weight, and higher-order modes on the accuracy of the numerical scheme. We demonstrate that, by applying the orthonormalization process, the stability and convergence of the approach are significantly improved.

Calculation of Lightning Induced Voltages on Overhead Lines Using an Analytical Fitting Representation of Electric Fields

The lightning electromagnetic field which illuminated the overhead line is the nonuniform field. When analyzing the lightning induced overvoltages on overhead lines by in frequency domain, to get an accurate response, the observation points of the horizontal electric field along the overhead line should be as many as possible. However, the expression of the horizontal electric field is formulated in the integral form and should be solved by the time-consuming numerical integration process at every observation point, which leads to a low calculation efficiency. This paper proposes an efficient method to avoid the numerical integration process at all the observation points and to enhance the calculation efficiency. In this method, the horizontal electric field at a few observation points along the line are calculated in the numerical way first, then the amplitudes and phases of the horizontal electric field along the line are fitted separately at every frequency with the fitting functions of the position $x$ along the line. The amplitudes are fitted by the Fourier series, and the phases are fitted by the polynomial functions. Based on the obtained analytical fitting representations, the transmission line response could be solved efficiently. The two different lightning electromagnetic pulse (LEMP) validation examples show the validity and general applicability of the proposed method.

GPU-acceleration of waveform relaxation methods for large differential systems

It is the purpose of this paper to provide an acceleration of waveform relaxation (WR) methods for the numerical solution of large systems of ordinary differential equations. The introduced technique is based on the employ of graphics processing units (GPUs) in order to speed-up the numerical integration process. A CUDA solver based on WR-Picard, WR-Jacobi and red-black WR-Gauss-Seidel iterations is presented and some numerical experiments realized on a multi-GPU machine are provided.

An Accurate Method for Real-Time Aircraft Dynamics Simulation Based on Predictor-Corrector Scheme

Real-time aircraft dynamics simulation requires very high accuracy and stability in the numerical integration process. Nonetheless, traditional multistep numerical methods cannot effectively meet the new requirements. Therefore, a novel real-time multistep method based on Predict-Evaluate-Correct scheme of three-step fourth-order method (RTPEC-34) is proposed and developed in this research to address the gap. In addition to the development of a highly accurate algorithm based on predictor-corrector, the contribution of this work also includes the analysis of truncation error for real-time problems. Moreover, the parameters for the RTPEC-34 method are optimized using intelligent optimization algorithms. The application and comparison of the optimization algorithms also lead to general guidelines for their applications in the development of improved multistep methods. Last but not least, theoretical analysis is also conducted on the stability of the proposed RTPEC-34 method, which is corroborated in simulation experiments and thus provides general guidelines for the evaluation of real-time numerical methods. The RTPEC-34 method is compared with other multistep algorithms using both numerical experiments and a real engineering example. As shown in the comparison, it achieves improved performance in terms of accuracy and stability and it is also a viable and efficient algorithm for real-time aircraft dynamics simulation.

Aeroelastic Analysis of Horizontal Axis Wind Turbine Based on Multi-body Model

The alternate loads like aerodynamic loads, inertial force and elastic force can lead to the coupled vibration of flexible wind rotor, tower and other components, and this coupled vibration can dramatically affects the aerodynamic characteristics and service life of wind turbine. The establishment of aeroelastic coupling mathematical model of horizontal axis wind turbine and its numerical integration method are studied. The super-element(SE) which can describe elastic deformation is used to distretize the flexible component into a series of rigid bodies connected by rotational joints with force element springs and dampers. The dynamic equations of a constrained multi-body system(MBS) of wind turbine is automatically set up and numerically solved through a designed simulation program based on the theory of dynamics of multi-body system and the modelling method of hybrid multi-body system(HMBS). The analysis of dynamic characteristics of the system is realized through Fourier spectrum analysis method. The natural frequencies and modal shapes of a 5 MW offshore wind turbine of United States National Renewable Energy Laboratory(NREL) are analyzed and the effectiveness of the program and the correctness of the modeling method are verified. Aerodynamic loads acting on each rigid body of blade under deformation state are calculated based the blade element momentum(BEM) theory, and the coupling between the structural and aerodynamic analysis is carried out in real time during the process of numerical integration. The results show that super-element can accurately describe the coupling among the aerodynamic loads, inertial loads and elastic loads of wind turbine. The program developed in this study is a useful analysis platform for the aeroelastic coupling, stability analysis and the design of control system of wind turbine.

Linear and nonlinear analysis of bond-slip models for interfaces between FRP composites and concrete

The paper analyses different analytical and numerical solutions for the debonding process of the FRP-to-concrete interface on shear tests with the FRP plate submitted to a tensile load in one of its ends. From the point of view of the state of the art, two different ways of finding the bond-slip curve from experiments are discussed and analysed. Essentially, three different linear bond-slip models, one exponential model and another power based function are employed in the numerical process. The results are analysed and compared. The differences found in the stress field along the interface, maximum load, maximum slip, ultimate slip, fracture energy and effective bond length are reported. The load-slip behaviour is also presented for the linear and non-linear models herein studied and the influence of the local bond-slip model on the debonding process is discussed. The numerical integration process used on the present study proved to be coherent with the analytical expressions determined for the linear bond-slip models and allowed to verify that maximum load transmittable to the FRP plate is influenced by the square root of the FRP stiffness and fracture energy even when nonlinear bond-slip models are assumed.

Chebyshev tau meshless method based on the highest derivative for fourth order equations

It is well known that the numerical integration process is much less sensitive than numerical differential process when dealing with the differential equations. After integration, accuracy is no longer limited by that of the slowly convergent series for the highest derivative, but only by that of the unknown function itself. In this paper, a Chebyshev tau meshless method based on the highest derivative (CTMMHD) is developed for fourth order equations on irregularly shaped domains with complex boundary conditions. The problem domain is embedded in a domain of regular shape. The integration and multiplication of Chebyshev expansions are given in matrix representations. Several numerical experiments including standard biharmonic problems, problems with variable coefficients and nonlinear problems are implemented to verify the high accuracy and efficiency of our method.

Integration Processes of Delay Differential Equation Based on Modified Laguerre Functions

We propose long‐time convergent numerical integration processes for delay differential equations. We first construct an integration process based on modified Laguerre functions. Then we establish its global convergence in certain weighted Sobolev space. The proposed numerical integration processes can also be used for systems of delay differential equations. We also developed a technique for refinement of modified Laguerre‐Radau interpolations. Lastly, numerical results demonstrate the spectral accuracy of the proposed method and coincide well with analysis.

Legendre-Gauss-Radau Collocation Method for Solving Initial Value Problems of First Order Ordinary Differential Equations

In this paper, we propose an efficient numerical integration process for initial value problems of first order ordinary differential equations, based on Legendre-Gauss-Radau interpolation, which is easy to be implemented and possesses the spectral accuracy. We also develop a multi-step version of this approach, which can be regarded as a specific implicit Legendre-Gauss-Radau Runge-Kutta method, with the global convergence and the spectral accuracy. Numerical results coincide well with the theoretical analysis and demonstrate the effectiveness of these approaches.

Development of a quadratic finite element formulation based on the XFEM and NURBS

AbstractThe FE‐simulation of inhomogeneous structures, such as composite materials, biological tissues or foams, requires the generation of respective finite element meshes. With increasing complexity of the inner architecture of such structures, this becomes a time‐consuming and laborious task. Additionally, the risk of forming bad‐shaped elements that may lead to ill‐conditioned numerical problems grows significantly. A solution to this problem provides the extended finite element method (XFEM). Thereby, the interface between different materials is represented by a local enrichment of the displacement approximation. As a consequence of this, the element boundary need not be aligned to the interface.In order to improve the accuracy of the interface approximation, the development of a plane element based on the XFEM and quadratic shape functions will be presented. This element allows for the description of curved material interfaces. The computation of the element stiffness matrix requires a numerical integration process that accounts for discontinuous fields. Regarding a linear element formulation, this can be achieved by an adapted triangulation of the element domain. However, in the case of a curved interface this solution is not applicable. Hence, non‐uniform rational B‐Spline (NURBS) surfaces are used to evaluate the integrals numerically.Finally, the results of different examples will show the general properties such as the accuracy of the numerical integration procedure and the convergence behavior of this element formulation. Copyright © 2011 John Wiley & Sons, Ltd.

ドメイン分割連成方式によるパワーエレクトロニクスシステムのマルチレート解析法

A power electronic system generally has multiple domains, and the rate of change of variables varies across the domains. Such systems are called multirate systems and they can be efficiently simulated by adopting numerical integration process according to the multirate characteristic. Two methods for performing the multirate analysis have been proposed. The first method is the circuit partitioning method in which a system is divided with large energy-storage elements into subcircuits which have their own optimum time step sizes for numerical integration. For example, a system in which converters and inverters are connected in cascade with large capacitors is divided into subcircuits and analyzed efficiently. The other method is the envelope-following (EF) method; in this method only selected voltage and current ripples are computed so that the envelopes corresponding to these ripples can be accurately determined. The number of skipped switch cycles can be adjusted optimally according to the variation rates of the envelopes. A new multirate method is proposed in this study. It is a combination of the two abovementioned methods, and therefore, it has the advantages of both the methods. First, the computational flow of the proposed method is described briefly. Then, precise numerical processes, for example, the selection of the step sizes of the fast and slow domains and the iteration correction taking into consideration the interactions between the domains, are described. Finally, a simulated motor drive system is investigated to verify the effectiveness of the proposed method.

Instability in the transformation between extensional and flexural modes in thin-walled cylindrical shells

With the inclusion of all terms up to the third order in the membrane strain to consider the geometric nonlinearity in deformation, a third-order implicit model and a fourth-order explicit model for the vibration transformation between extensional and flexural modes in thin-walled cylindrical shells are established and solved numerically. Numerical instability is observed in numerical solutions based on the explicit model. It is found that such numerical instability does not result from the accumulated numerical errors in the numerical integration process, but from the neglecting of higher order terms in the formulation of the problem. With the inclusion of all terms up to the fourth order in the strain energy, the explicit model can predict a stable vibration history with periodic mode transformations between the two modes. The same 2:1 internal resonance of the vibration mode transformation is predicted by both models. As flexural stress growth is concerned, the two models matches very well during the first group of peaks but lag in phase gradually appears in later group of stress peaks based on the implicit model. This is understood to be resulting from the limited number of terms included in the series expansions based on the explicit model, which allows the most likely excited flexural mode get a larger share of energy transferred from the principle mode and as a result flexural stress arrives at peaks earlier based on the explicit model.

Computationally efficient signal modeling for vertical scanning interferometry

This paper presents a computationally efficient model to simulate the interference signal of vertical scanning interferometry. Existing models are either oversimplified or computationally intensive. Our model incorporates the geometric and spectral effects on vertical scanning interferometry, but removes the computationally intensive numerical integration process by modeling the light spectrum as a sum of piecewise cosine functions. Compared to direct numerical integration of the generalized model, the computational time (for an interference signal) of the proposed model is 256,800 times faster. To verify the accuracy of the proposed model, we simulate the interference signal of a phosphor-based LED, and verify our result with experimental data and a computationally intensive counterpart. Other than reduced computational time, the elementary form of an interference signal derived in this paper will facilitate future work.

Evaluating methods for measuring the mean soil temperature

Measurements of temperatures that are averaged or integrated over long time spans are needed for a variety of purposes such as soil resource inventory and climate studies. Mean temperatures usually are obtained in a two-step process of observation and numerical integration of periodic values. Pallmann introduced a technique using a temperature driven chemical reaction to integrate temperature over a period time. In this research, we illustrated the field application of the Pallmann method to measure mean soil temperature in a long-term climate change investigation project. The accuracy and consistency of the Pallmann method were evaluated by comparing two other temperature measurement methods: thermistor sensor and diffusion-cell method. The consistency of the Pallmann method is quite good. On average, the mean annual soil temperatures measured by the Pallmann method were 0.27 °C lower than those measured by the diffusion-cell method and were 0.64 °C higher than those measured by the thermistor sensor method. While statistical analysis showed that there was significant difference between the means of these three methods, the Pallmann method may better reflect the real ecological processes in soil in comparison with the arithmetic-mean temperature measured by the thermal sensors. It is an ideal method for studying the spatial variation in soil temperature and long-term climate changes.

Development of a quadratic element formulation based on the X‐FEM and NURBS surfaces

AbstractThe FE‐simulation of inhomogeneous structures such as composite materials, biological tissues or foams requires the generation of respective finite element meshes. The Extended Finite Element Method (X‐FEM) allows for an efficient modelling technique where the element boundary needs not to be aligned to the interfaces. In this paper, a new plane element based on the X‐FEM and quadratic shape functions will be presented. This element allows for the description of curved material interfaces inside the element domain. The computation of the element stiffness matrix requires a numerical integration process which accounts for the discontinuous fields. In the case of a curved interface, the triangulation of the element domain as applied for linear element formulations is inappropriate. Hence, Non‐Uniform Rational B‐Spline (NURBS) surfaces are used to evaluate the integrals numerically. Finally, the results of different examples will show the properties and the quality of this element formulation. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Coupling upscaling finite element method for consolidation analysis of heterogeneous saturated porous media

The coupling upscaling finite element method is developed for solving the coupling problems of deformation and consolidation of heterogeneous saturated porous media under external loading conditions. The method couples two kinds of fully developed methodologies together, i.e., the numerical techniques developed for calculating the apparent and effective physical properties of the heterogeneous media and the upscaling techniques developed for simulating the fluid flow and mass transport properties in heterogeneous porous media. Equivalent permeability tensors and equivalent elastic modulus tensors are calculated for every coarse grid block in the coarse-scale model of the heterogeneous saturated porous media. Moreover, an oversampling technique is introduced to improve the calculation accuracy of the equivalent elastic modulus tensors. A numerical integration process is performed over the fine mesh within every coarse grid element to capture the small scale information induced by non-uniform scalar field properties such as density, compressibility, etc. Numerical experiments are carried out to examine the accuracy of the developed method. It shows that the numerical results obtained by the coupling upscaling finite element method on the coarse-scale models fit fairly well with the reference solutions obtained by traditional finite element method on the fine-scale models. Moreover, this method gets more accurate coarse-scale results than the previously developed coupling multiscale finite element method for solving this kind of coupling problems though it cannot recover the fine-scale solutions. At the same time, the method developed reduces dramatically the computing effort in both CPU time and memory for solving the transient problems, and therefore more large and computational-demanding coupling problems can be solved by computers.

Influence of trans-neptunian objects on motion of major planets and limitation on the total TNO mass from planet and spacecraft ranging

AbstractPerturbations from asteroids and Trans-Neptunian Objects affect significantly on the orbits of planets and should be taken into account when high-accuracy planetary ephemerides are constructed. On the other hand, from an analysis of motion of the major planets by processing of precise measurements of spacecraft a limitation on the total TNO mass may be obtained. To estimate influence of TNO on motion of planets the largest 21 TNO have been included into the process of simultaneous numerical integration, and positions of planets obtained with taking for TNO have been compared with positions of planets of numerical EPM ephemeris of IAA RAS constructed without these objects. The perturbations of other TNO have been modeled by the perturbation from a circular ring having a radius 43 AU and different masses. It has been shown that all the test masses of the TNO ring except the minimum mass (5.26⋅10−8M⊙) are too large and make the data residuals worse. Thus, the upper limit of the total mass of all TNO including Pluto, the 21 largest TNO and the TNO ring (with the 43 AU radius) should not exceed 8.04⋅10−8M⊙.

Legendre–Gauss collocation methods for ordinary differential equations

In this paper, we propose two efficient numerical integration processes for initial value problems of ordinary differential equations. The first algorithm is the Legendre–Gauss collocation method, which is easy to be implemented and possesses the spectral accuracy. The second algorithm is a mixture of the collocation method coupled with domain decomposition, which can be regarded as a specific implicit Legendre–Gauss Runge–Kutta method, with the global convergence and the spectral accuracy. Numerical results demonstrate the spectral accuracy of these approaches and coincide well with theoretical analysis.