The Hamilton-Jacobi method applied to the low-thrust trajectory problem.
Spacecraft optimum low thrust trajectory analyzed by Hamilton-Jacobi perturbation theory, obtaining canonic constants of motion
Miner, W. E.↗
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Results for “Jacobi Method”
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Jacobi-Integral Method For Two-Body Problem
Jacobi-integral method enables efficient, accurate computation of trajectory of natural satellite or spacecraft perturbed by component of gravitational potential depending explicitly on both position and time. Instead of total energy, Jacobi integral, which is energylike constant of motion in this case, embedded in Newtonian differential equations of motion. Trajectories computed in fewer steps. With modifications, applicable to such terrestrial problems as motions of rotors and of beams of electrically charged particles in changing electrical and magnetic fields.
Bond, Victor R.↗
Numerical algorithms for finite element computations on arrays of microprocessors
The development of a multicolored successive over relaxation (SOR) program for the finite element machine is discussed. The multicolored SOR method uses a generalization of the classical Red/Black grid point ordering for the SOR method. These multicolored orderings have the advantage of allowing the SOR method to be implemented as a Jacobi method, which is ideal for arrays of processors, but still enjoy the greater rate of convergence of the SOR method. The program solves a general second order self adjoint elliptic problem on a square region with Dirichlet boundary conditions, discretized by quadratic elements on triangular regions. For this general problem and discretization, six colors are necessary for the multicolored method to operate efficiently. The specific problem that was solved using the six color program was Poisson's equation; for Poisson's equation, three colors are necessary but six may be used. In general, the number of colors needed is a function of the differential equation, the region and boundary conditions, and the particular finite element used for the discretization.
Ortega, J. M.↗
Scalar-tensor theory in canonical form. II - Exact solutions for a static spherically symmetric charged mass distribution
Static spherically symmetric charged mass distribution exact solutions derived for electrons in scalar tensor theory of gravity by Hamilton-Jacobi method
Toton, E. T.↗
Clock synchronization by accelerated observers - Metric construction for arbitrary congruences of world lines
Clock synchronization in an arbitrarily accelerated observer congruence is considered. A general solution is obtained that maintains the isotropy and coordinate independence of the one-way speed of light. Attention is also given to various particular cases including, rotating disk congruence or ring congruence. An explicit, congruence-based spacetime metric is constructed according to Einstein's clock synchronization procedure and the equation for the geodesics of the space-time was derived using Hamilton-Jacobi method. The application of interferometric techniques (absolute phase radio interferometry, VLBI) to the detection of the 'global Sagnac effect' is also discussed.
Henriksen, R. N.↗
Time integration algorithms for the two-dimensional Euler equations on unstructured meshes
Explicit and implicit time integration algorithms for the two-dimensional Euler equations on unstructured grids are presented. Both cell-centered and cell-vertex finite volume upwind schemes utilizing Roe's approximate Riemann solver are developed. For the cell-vertex scheme, a four-stage Runge-Kutta time integration, a fourstage Runge-Kutta time integration with implicit residual averaging, a point Jacobi method, a symmetric point Gauss-Seidel method and two methods utilizing preconditioned sparse matrix solvers are presented. For the cell-centered scheme, a Runge-Kutta scheme, an implicit tridiagonal relaxation scheme modeled after line Gauss-Seidel, a fully implicit lower-upper (LU) decomposition, and a hybrid scheme utilizing both Runge-Kutta and LU methods are presented. A reverse Cuthill-McKee renumbering scheme is employed for the direct solver to decrease CPU time by reducing the fill of the Jacobian matrix. A comparison of the various time integration schemes is made for both first-order and higher order accurate solutions using several mesh sizes, higher order accuracy is achieved by using multidimensional monotone linear reconstruction procedures. The results obtained for a transonic flow over a circular arc suggest that the preconditioned sparse matrix solvers perform better than the other methods as the number of elements in the mesh increases.
Slack, David C.↗
A Discontinuous Galerkin Finite Element Method for Hamilton-Jacobi Equations
In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the Runge-Kutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method.
Hu, Changqing↗
Matrix methods for calculating zeros, coefficients, Christoffel numbers, and derivatives of some orthogonal polynomials
Jacobi matrix method for calculating zeros, coefficients, Christoffel numbers, and derivatives of orthogonal polynomials
MATRIX ALGEBRA↗
A multigrid method for steady Euler equations on unstructured adaptive grids
A flux-difference splitting type algorithm is formulated for the steady Euler equations on unstructured grids. The polynomial flux-difference splitting technique is used. A vertex-centered finite volume method is employed on a triangular mesh. The multigrid method is in defect-correction form. A relaxation procedure with a first order accurate inner iteration and a second-order correction performed only on the finest grid, is used. A multi-stage Jacobi relaxation method is employed as a smoother. Since the grid is unstructured a Jacobi type is chosen. The multi-staging is necessary to provide sufficient smoothing properties. The domain is discretized using a Delaunay triangular mesh generator. Three grids with more or less uniform distribution of nodes but with different resolution are generated by successive refinement of the coarsest grid. Nodes of coarser grids appear in the finer grids. The multigrid method is started on these grids. As soon as the residual drops below a threshold value, an adaptive refinement is started. The solution on the adaptively refined grid is accelerated by a multigrid procedure. The coarser multigrid grids are generated by successive coarsening through point removement. The adaption cycle is repeated a few times. Results are given for the transonic flow over a NACA-0012 airfoil.
Riemslagh, Kris↗
Applications of partial orderings to the study of positive definiteness, monotonicity, and convergence of iterative methods for linear systems.
Most of the known results concerning convergence of iterative methods for solving linear systems involve either positive definiteness or monotonicity. In this paper a new concept, called K-semipositivity, is introduced, which provides a link between convergence theory, monotonicity, and positive definiteness. By using this concept, together with partial orderings in Euclidean n-space, several new convergence theorems are proved. Application to Jacobi's methods and the theory of regular splittings shows the usefulness of these new results.
Vandergraft, J. S.↗
A block iterative LU solver for weakly coupled linear systems
A hybrid technique, called the block iterative LU solver, is proposed for solving the linear equations resulting from a finite element numerical analysis of certain fluid dynamics problems where the equations are weakly coupled between distinct sets of variables. Either the block Jacobi iterative method or the block Gauss-Seidel iterative solver is combined with LU decomposition.
Cooke, C. H.↗
The effect of gravity level on the average primary dendritic spacing of a directionally solidified superalloy
The effect of alternating low (0.01 g) and high (1.8 g) gravity force on the primary spacings in the dendrite structure in a directionally solidified Ni-based superalloy (PWA 1480, containing 5 pct Co, 10 pct Cr, 4 pct W, 12 pct Ta, 5 pct Al, 1.5 pct Ti, and the balance Ni) was investigated using samples solidified in a directional solidification furnace aboard the NASA KC-135 aircraft that made a series of low-g parabolas. The cross-section slices for each growth rate were polished and etched with Kallings II, and the primary dendritic arm spacings were measured using the method of Jacobi and Schwerdtfeger (1976). The arm spacings were found to fluctuate with gravity force, increasing as the gravity level decreased, and growing finer as gravity increased.
Mccay, M. H.↗
Acoustic modes in fluid networks
Pressure and flow rate eigenvalue problems for one-dimensional flow of a fluid in a network of pipes are derived from the familiar transmission line equations. These equations are linearized by assuming small velocity and pressure oscillations about mean flow conditions. It is shown that the flow rate eigenvalues are the same as the pressure eigenvalues and the relationship between line pressure modes and flow rate modes is established. A volume at the end of each branch is employed which allows any combination of boundary conditions, from open to closed, to be used. The Jacobi iterative method is used to compute undamped natural frequencies and associated pressure/flow modes. Several numerical examples are presented which include acoustic modes for the Helium Supply System of the Space Shuttle Orbiter Main Propulsion System. It should be noted that the method presented herein can be applied to any one-dimensional acoustic system involving an arbitrary number of branches.
Michalopoulos, C. D.↗
Phase shifts and the quantum-mechanical hamilton-jacobi equation.
Phase shift method using integration of quantum- mechanical Hamilton-Jacobi equation applied to particles interacting through Lennard-Jones potential
PHASE SHIFT↗
Parallel solution of finite element equations
The paper examines several parallel processing solution algorithms for finite element equations arising in linear equilibrium problems. Two basic groups of algorithms, direct and iterative, are investigated with respect to a number of parallel computer architectures and associated selection criteria. The direct algorithms include: LR-Gauss, Crout, Cholesky, Cyclic Reduction and WZ-factorization. The iterative methods examined are: Accelerated Gauss-Seidel, Surrogate Stiffness, Jacobi, Series Expansion, and Energy Monte Carlo. For real-time applications, where the object is to minimize the execution time, Cyclic Reduction appears to be best suited. This assumes a computer with an unlimited number of parallel processors. However, for computers with a limited number of parallel processors that must be used efficiently, both Gauss factorization and Jacobi-like iterative methods rank favorably.
Salama, M.↗
Discrete analog of an iterative method for inverse eigenvalue problems for Jacobi matrices
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Barcilon, V.↗
Experience in grid optimization
Two optimization methods for solving a variational problem in grid generation are described and evaluated. The smoothness, cell volumes, and orthogonality of the variational integrals are examined. The Jacobi-Newton iterative method is compared to the Fletcher-Reeves conjugate gradient method. It is observed that a combination of the Jacobi-Newton iteration and the direct solution of the variational problem produces an algorithm which is easy to program and requires less storage and computer time/iteration than the conjugate gradient method.
Mastin, C. W.↗
Comparing direct and iterative equation solvers in a large structural analysis software system
Two direct Choleski equation solvers and two iterative preconditioned conjugate gradient (PCG) equation solvers used in a large structural analysis software system are described. The two direct solvers are implementations of the Choleski method for variable-band matrix storage and sparse matrix storage. The two iterative PCG solvers include the Jacobi conjugate gradient method and an incomplete Choleski conjugate gradient method. The performance of the direct and iterative solvers is compared by solving several representative structural analysis problems. Some key factors affecting the performance of the iterative solvers relative to the direct solvers are identified.
Poole, E. L.↗