The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
John L. Junkins - One of the best experts on this subject based on the ideXlab platform.
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Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration
The Journal of the Astronautical Sciences, 2017Co-Authors: Robyn M. Woollands, Julie L. Read, Austin B. Probe, John L. JunkinsAbstract:We present a new method for solving the multiple revolution perturbed Lambert problem using the method of particular solutions and modified Chebyshev-Picard Iteration . The method of particular solutions differs from the well-known Newton-shooting method in that integration of the state transition matrix (36 additional differential equations) is not required, and instead it makes use of a reference trajectory and a set of n particular solutions. Any numerical integrator can be used for solving two-point boundary problems with the method of particular solutions , however we show that using modified Chebyshev-Picard Iteration affords an avenue for increased efficiency that is not available with other step-by-step integrators. We take advantage of the path approximation nature of modified Chebyshev-Picard Iteration (nodes iteratively converge to fixed points in space) and utilize a variable fidelity force model for propagating the reference trajectory. Remarkably, we demonstrate that computing the particular solutions with only low fidelity function evaluations greatly increases the efficiency of the algorithm while maintaining machine precision accuracy. Our study reveals that solving the perturbed Lambert’s problem using the method of particular solutions with modified Chebyshev-Picard Iteration is about an order of magnitude faster compared with the classical shooting method and a tenth-twelfth order Runge-Kutta integrator. It is well known that the solution to Lambert’s problem over multiple revolutions is not unique and to ensure that all possible solutions are considered we make use of a reliable preexisting Keplerian Lambert solver to warm start our perturbed algorithm.
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Application of Modified Chebyshev Picard Iteration to Differential Correction for Improved Robustness and Computation Time
The Journal of the Astronautical Sciences, 2017Co-Authors: Travis Swenson, Robyn Woollands, John L. JunkinsAbstract:A novel application of Modified Chebyshev Picard Iteration (MCPI) to differential correction is presented. By leveraging the Chebyshev basis functions of MCPI, interpolation in 1 dimension may be used to target plane crossing events, instead of integrating the 42 dimensional variational equation required for standard step integrators. This results in dramatically improved performance over traditional differential correctors. MCPI was tested against the Runge-Kutta 7/8 integrator on over 45,000 halo orbits in three different three-body problems, and was found to be up to an order of magnitude faster, while simultaneously increasing robustness.
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new solutions for the perturbed lambert problem using regularization and Picard Iteration
Journal of Guidance Control and Dynamics, 2015Co-Authors: Robyn M. Woollands, Ahmad Bani Younes, John L. JunkinsAbstract:A new approach for solving two-point boundary value problems and initial value problems using the Kustaanheimo–Stiefel transformation and Modified Chebyshev–Picard Iteration is presented. The first contribution is the development of an analytical solution to the elliptic Keplerian Lambert problem based on Kustaanheimo–Stiefel regularization. This transforms the nonlinear three-dimensional orbit equations of motion into four linear oscillators. The second contribution solves the elliptic Keplerian two-point boundary value problem and initial value problem using the Kustaanheimo–Stiefel transformation and Picard Iteration. The Picard sequence of trajectories represents a contraction mapping that converges to a unique solution over a finite domain. Solving the Keplerian two-point boundary value problem in Kustaanheimo–Stiefel variables increases the Picard domain of convergence from about one-third of an orbit (Cartesian variables) to over 95% of an orbit (Kustaanheimo–Stiefel variables). These increases in ...
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State Transition Matrix for Perturbed Orbital Motion Using Modified Chebyshev Picard Iteration
The Journal of the Astronautical Sciences, 2015Co-Authors: Julie L. Read, Ahmad Bani Younes, Brent Macomber, James Turner, John L. JunkinsAbstract:The Modified Chebyshev Picard Iteration (MCPI) method has recently proven to be highly efficient for a given accuracy compared to several commonly adopted numerical integration methods, as a means to solve for perturbed orbital motion. This method utilizes Picard Iteration, which generates a sequence of path approximations, and Chebyshev Polynomials, which are orthogonal and also enable both efficient and accurate function approximation. The nodes consistent with discrete Chebyshev orthogonality are generated using cosine sampling; this strategy also reduces the Runge effect and as a consequence of orthogonality, there is no matrix inversion required to find the basis function coefficients. The MCPI algorithms considered herein are parallel-structured so that they are immediately well-suited for massively parallel implementation with additional speedup. MCPI has a wide range of applications beyond ephemeris propagation, including the propagation of the State Transition Matrix (STM) for perturbed two-body motion. A solution is achieved for a spherical harmonic series representation of earth gravity (EGM2008), although the methodology is suitable for application to any gravity model. Included in this representation the normalized, Associated Legendre Functions are given and verified numerically. Modifications of the classical algorithm techniques, such as rewriting the STM equations in a second-order cascade formulation, gives rise to additional speedup. Timing results for the baseline formulation and this second-order formulation are given.
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multisegment scheme applications to modified chebyshev Picard Iteration method for highly elliptical orbits
Mathematical Problems in Engineering, 2015Co-Authors: Donghoon Kim, John L. Junkins, James D TurnerAbstract:A modified Chebyshev Picard Iteration method is proposed for solving orbit propagation initial/boundary value problems. Cosine sampling techniques, known as Chebyshev-Gauss-Lobatto (CGL) nodes, are used to reduce Runge’s phenomenon that plagues many series approximations. The key benefit of using the CGL data sampling is that the nodal points are distributed nonuniformly, with dense sampling at the beginning and ending times. This problem can be addressed by a nonlinear time transformation and/or by utilizing multiple time segments over an orbit. This paper suggests a method, called a multisegment method, to obtain accurate solutions overall regardless of initial states and albeit eccentricity by dividing the given orbit into two or more segments based on the true anomaly.
Xuechuan Wang - One of the best experts on this subject based on the ideXlab platform.
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application of modified chebyshev Picard Iteration to the relative orbital dynamics problem
2021 International Bhurban Conference on Applied Sciences and Technologies (IBCAST), 2021Co-Authors: Ali Imran, Xuechuan Wang, Faizan Sikandar Wains, Yue XiaokuiAbstract:In this paper we apply the Modified Chebyshev Picard Iteration Method, which uses Chebyshev Polynomials as basis functions to the Orbital Relative Dynamics Problem. We use the mathematical properties of the Chebyshev polynomials to speed up the integration process by parallel problem solving. In this paper we will only apply this method to the Clohessy-Wiltshire model of satellite relative motion, as their is a analytical solution available so we can easily quantify our results in comparison to the analytical method and other numerical methods like ODE45 and ODE113.
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bifurcation chaos in nonlinear structural dynamics novel highly efficient optimal feedback accelerated Picard Iteration algorithms
Communications in Nonlinear Science and Numerical Simulation, 2018Co-Authors: Xuechuan Wang, Weicheng Pei, S N AtluriAbstract:Abstract A new class of algorithms for solving nonlinear structural dynamical problems are derived in the present paper, as being based on optimal-feedback-accelerated Picard Iteration, wherein the solution vectors for the displacements and velocities at any time t in a finitely large time interval t i ≤ t ≤ t i + 1 are corrected by a weighted (with a matrix λ) integral of the error from ti to t. We present 3 approximations to solve the Euler-Lagrange equations for the optimal weighting functions λ; thus we present 3 algorithms denoted as Optimal-Feedback-Accelerated Picard Iteration (OFAPI) algorithms-1, 2, 3. The interval ( t i + 1 − t i ) in the 3 OFAPI algorithms can be several hundred times larger than the increment (Δt) required in the finite difference based implicit or explicit methods, for the same stability and accuracy. Moreover, the OFAPI algorithms-2, 3 do not require the inversion of the tangent stiffness matrix, as is required in finite difference based implicit methods. It is found that OFAPI algorithms-1, 2, 3 (especially OFAPI algorithm-2) require several orders of magnitude of less computational time than the currently popular implicit and explicit finite difference methods, and provide better accuracy and convergence.
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feedback accelerated Picard Iteration for orbit propagation and lambert s problem
Journal of Guidance Control and Dynamics, 2017Co-Authors: Xuechuan Wang, Satya N AtluriAbstract:This paper presents a new feedback-accelerated Picard Iteration method for solving long-term orbit-propagation problems and perturbed Lambert’s problems. This method is developed by combining the c...
Th M Van Genuchten - One of the best experts on this subject based on the ideXlab platform.
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solution of the nonlinear transport equation using modified Picard Iteration
Advances in Water Resources, 1998Co-Authors: K Huang, Binayak P Mohanty, Feike J Leij, Th M Van GenuchtenAbstract:Abstract The transport and fate of reactive chemicals in groundwater is governed by equations which are often difficult to solve due to the nonlinear relationship between the solute concentrations for the liquid and solid phases. The nonlinearity may cause mass balance errors during the numerical simulation in addition to numerical errors for linear transport system. We have generalized the modified Picard Iteration algorithm of Celia et al.5 for unsaturated flow to solve the nonlinear transport equation. Written in a ‘mixed-form’ formulation, the total solute concentration is expanded in a Taylor series with respect to the solution concentration to linearize the transport equation, which is then solved with a conventional finite element method. Numerical results of this mixed-form algorithm are compared with those obtained with the concentration-based scheme using conventional Picard Iteration. In general, the new solver resulted in negligible mass balance errors (
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a new convergence criterion for the modified Picard Iteration method to solve the variably saturated flow equation
Journal of Hydrology, 1996Co-Authors: K Huang, Binayak P Mohanty, Th M Van GenuchtenAbstract:Abstract Solutions of the Richards equation for water flow in variably saturated porous media are increasingly being used in water resources evaluation and environmental management. Besides the accuracy of solution, also of concern is the required computational effort, especially when highly nonlinear soil hydraulic properties and dry initial conditions are involved. In this paper we evaluate the performance of different convergence criteria when the modified Picard Iteration method is used for solving the mixed-form Richards equation. Results are compared in terms of computer processing (CPU) time and number of Iterations. A new nonlinear convergence criterion derived using a Taylor series expansion of the water content was implemented in the mixed-form numerical algorithm. The computational efficiency of the new criterion was evaluated against two widely used convergence criteria for different soil types, boundary conditions, initial conditions, and layered soils. Whereas all three criteria produced nearly identical results in terms of calculated water content, pressure head, and waser flux distributions, all with negligible mass balance errors, the required CPU times were significantly different. In general, the new nonlinear convergence criterion was found to be computationally much more efficient than the other two criteria. The new criterion was also more robust (i.e. the solution remained convergent) for highly nonlinear flow problems for which the other two convergence criteria failed. Results of this study indicate that the new convergence criterion, when implemented in the modified Picard solution of the mixed-form Richards equation, produces a very efficient and accurate method for simulating variably saturated water flow in soils.
B E Rhoades - One of the best experts on this subject based on the ideXlab platform.
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stability of Picard Iteration in metric spaces
Nonlinear functional analysis and applications, 2013Co-Authors: Yuan Qing, B E RhoadesAbstract:In this paper we establish a general result for the stability of Picard Iteration. Several theorems in the literature are obtained as special cases.
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on stability of Picard Iteration in cone metric spaces
Fixed Point Theory and Applications, 2009Co-Authors: Mehdi Asadi, Hossein Soleimani, S M Vaezpour, B E RhoadesAbstract:The aim of this work is to investigate the -stability of Picard's Iteration procedures in cone metric spaces and give an application.
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t stability of Picard Iteration in metric spaces
Fixed Point Theory and Applications, 2008Co-Authors: Yuan Qing, B E RhoadesAbstract:We establish a general result for the stability of Picard's Iteration. Several theorems in the literature are obtained as special cases.
K Huang - One of the best experts on this subject based on the ideXlab platform.
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solution of the nonlinear transport equation using modified Picard Iteration
Advances in Water Resources, 1998Co-Authors: K Huang, Binayak P Mohanty, Feike J Leij, Th M Van GenuchtenAbstract:Abstract The transport and fate of reactive chemicals in groundwater is governed by equations which are often difficult to solve due to the nonlinear relationship between the solute concentrations for the liquid and solid phases. The nonlinearity may cause mass balance errors during the numerical simulation in addition to numerical errors for linear transport system. We have generalized the modified Picard Iteration algorithm of Celia et al.5 for unsaturated flow to solve the nonlinear transport equation. Written in a ‘mixed-form’ formulation, the total solute concentration is expanded in a Taylor series with respect to the solution concentration to linearize the transport equation, which is then solved with a conventional finite element method. Numerical results of this mixed-form algorithm are compared with those obtained with the concentration-based scheme using conventional Picard Iteration. In general, the new solver resulted in negligible mass balance errors (
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a new convergence criterion for the modified Picard Iteration method to solve the variably saturated flow equation
Journal of Hydrology, 1996Co-Authors: K Huang, Binayak P Mohanty, Th M Van GenuchtenAbstract:Abstract Solutions of the Richards equation for water flow in variably saturated porous media are increasingly being used in water resources evaluation and environmental management. Besides the accuracy of solution, also of concern is the required computational effort, especially when highly nonlinear soil hydraulic properties and dry initial conditions are involved. In this paper we evaluate the performance of different convergence criteria when the modified Picard Iteration method is used for solving the mixed-form Richards equation. Results are compared in terms of computer processing (CPU) time and number of Iterations. A new nonlinear convergence criterion derived using a Taylor series expansion of the water content was implemented in the mixed-form numerical algorithm. The computational efficiency of the new criterion was evaluated against two widely used convergence criteria for different soil types, boundary conditions, initial conditions, and layered soils. Whereas all three criteria produced nearly identical results in terms of calculated water content, pressure head, and waser flux distributions, all with negligible mass balance errors, the required CPU times were significantly different. In general, the new nonlinear convergence criterion was found to be computationally much more efficient than the other two criteria. The new criterion was also more robust (i.e. the solution remained convergent) for highly nonlinear flow problems for which the other two convergence criteria failed. Results of this study indicate that the new convergence criterion, when implemented in the modified Picard solution of the mixed-form Richards equation, produces a very efficient and accurate method for simulating variably saturated water flow in soils.