Search results for "shooting method"
showing 10 items of 18 documents
Numerical solution of a class of nonlinear boundary value problems for analytic functions
1982
We analyse a numerical method for solving a nonlinear parameter-dependent boundary value problem for an analytic function on an annulus. The analytic function to be determined is expanded into its Laurent series. For the expansion coefficients we obtain an operator equation exhibiting bifurcation from a simple eigenvalue. We introduce a Galerkin approximation and analyse its convergence. A prominent problem falling into the class treated here is the computation of gravity waves of permanent type in a fluid. We present numerical examples for this case.
Shooting methods for 1D steady-state free boundary problems
1993
AbstractIn this note, we present two numerical methods based on shooting methods to solve steady-state diffusion-absorption models.
Parallel fictitious domain method for a non‐linear elliptic neumann boundary value problem
1999
Parallelization of the algebraic fictitious domain method is considered for solving Neumann boundary value problems with variable coefficients. The resulting method is applied to the parallel solution of the subsonic full potential flow problem which is linearized by the Newton method. Good scalability of the method is demonstrated on a Cray T3E distributed memory parallel computer using MPI in communication. Copyright © 1999 John Wiley & Sons, Ltd.
Efficient finite difference formulation of a geometrically nonlinear beam element
2021
The article is focused on a two-dimensional geometrically nonlinear formulation of a Bernoulli beam element that can accommodate arbitrarily large rotations of cross sections. The formulation is based on the integrated form of equilibrium equations, which are combined with the kinematic equations and generalized material equations, leading to a set of three first-order differential equations. These equations are then discretized by finite differences and the boundary value problem is converted into an initial value problem using a technique inspired by the shooting method. Accuracy of the numerical approximation is conveniently increased by refining the integration scheme on the element lev…
Numerische Behandlung von Verzweigungsproblemen bei gew�hnlichen Differentialgleichungen
1979
We present a new method for the numerical solution of bifurcation problems for ordinary differential equations. It is based on a modification of the classical Ljapunov-Schmidt-theory. We transform the problem of determining the nontrivial branch bifurcating from the trivial solution into the problem of solving regular nonlinear boundary value problems, which can be treated numerically by standard methods (multiple shooting, difference methods).
Geometric optimal control : homotopic methods and applications
2012
This work is about geometric optimal control applied to celestial and quantum mechanics. We first dealt with the minimum fuel consumption problem of transfering a satellite around the Earth. This brought to the creation of the code HamPath which permits first of all to solve optimal control problem for which the command law is smooth. It is based on the Pontryagin Maximum Principle (PMP) and on the notion of conjugate point. This program combines shooting method, differential homotopic methods and tools to compute second order optimality conditions. Then we are interested in quantum control. We study first a system which consists in two different particles of spin 1/2 having two different r…
3D Geosynchronous Transfer of a Satellite: Continuation on the Thrust
2003
The minimum-time transfer of a satellite from a low and eccentric initial orbit toward a high geostationary orbit is considered. This study is preliminary to the analysis of similar transfer cases with more complicated performance indexes (maximization of payload, for instance). The orbital inclination of the spacecraft is taken into account (3D model), and the thrust available is assumed to be very small (e.g. 0.3 Newton for an initial mass of 1500 kg). For this reason, many revolutions are required to achieve the transfer and the problem becomes very oscillatory. In order to solve it numerically, an optimal control model is investigated and a homotopic procedure is introduced, namely cont…
Optimal control of the atmospheric arc of a space shuttle and numerical simulations with multiple-shooting method
2005
This article, continuation of previous works, presents the applications of geometric optimal control theory to the analysis of the Earth re-entry problem for a space shuttle where the control is the angle of bank, the cost is the total amount of thermal flux, and the system is subject to state constraints on the thermal flux, the normal acceleration and the dynamic pressure. Our analysis is based on the evaluation of the reachable set using the maximum principle and direct computations with the boundary conditions according to the CNES research project\footnote{The project is partially supported by the Centre National d'Etude Spatiales.}. The optimal solution is approximated by a concatenat…
A Multiplicity result for a class of strongly indefinite asymptotically linear second order systems
2010
We prove a multiplicity result for a class of strongly indefinite nonlinear second order asymptotically linear systems with Dirichlet boundary conditions. The key idea for the proof is to bring together the classical shooting method and the Maslov index of the linear Hamiltonian systems associated to the asymptotic limits of the given nonlinearity.
BEAM ELEMENT UNDER FINITE ROTATIONS
2021
The present work focuses on the 2-D formulation of a nonlinear beam model for slender structures that can exhibit large rotations of the cross sections while remaining in the small-strain regime. Bernoulli-Euler hypothesis that plane sections remain plane and perpendicular to the deformed beam centerline is combined with a linear elastic stress-strain law. The formulation is based on the integrated form of equilibrium equations and leads to a set of three first-order differential equations for the displacements and rotation, which are numerically integrated using a special version of the shooting method. The element has been implemented into an open-source finite element code to ease comput…