On the use of approximate analytical solutions in solving optimum trajectory problems
Method for numerical solutions of nonlinear optimum trajectory problems using approximate analytic method
APPROXIMATION METHOD↗
SEARCH · Engineering Papers
Results for “Nonlinear problems”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Program for the solution of multipoint boundary value problems of quasilinear differential equations
Linear equations are solved by a method of superposition of solutions of a sequence of initial value problems. For nonlinear equations and/or boundary conditions, the solution is iterative and in each iteration a problem like the linear case is solved. A simple Taylor series expansion is used for the linearization of both nonlinear equations and nonlinear boundary conditions. The perturbation method of solution is used in preference to quasilinearization because of programming ease, and smaller storage requirements; and experiments indicate that the desired convergence properties exist although no proof or convergence is given.
Source record↗
Solution techniques for incompressible flow problems
A three-step Petrov-Galerkin (PG)/operator spliting scheme for the time-dependent incompressible Navier-Stokes equations is proposed. Each time step is split into two Stokes problems and one nonlinear convection-diffusion problem. Using a PG technique on the two outer Stokes problems ensures a stable scheme despite equal-order interpolation, while using a streamline upwind PG scheme on the inner convection-diffusion problem ensures a numerically stable solution at high Reynolds numbers. Numerical tests of this method have been carried out.
Tezduyar, T. E.↗
A nonlinear, discrete-vortex-perturbation method for unsteady lifting-surface problems with edge separations
The nonlinear-discrete vortex method is coupled with a perturbation method to solve the problem of a rectangular wing with small oscillation about high angles of attack. The solution of the problem is based on decoupling the steady and unsteady effects. The steady part of the problem is a nonlinear one and is solved by the nonlinear-discrete vortex method. The unsteady part of the problem is a linear one and is solved directly without any iteration. So far, the developed method is restricted to flat rectangular surfaces with pitching oscillations. Total and distributed loads of several rectangular wings are presented as numerical results.
Kandil, O. A.↗
Comparison of some optimal control methods for the design of turbine blades
This paper attempts a comparative study of some numerical methods for the optimal control design of turbine blades whose vibration characteristics are approximated by Timoshenko beam idealizations with shear and incorporating simple boundary conditions. The blade was synthesized using the following methods: (1) conjugate gradient minimization of the system Hamiltonian in function space incorporating penalty function transformations, (2) projection operator methods in a function space which includes the frequencies of vibration and the control function, (3) epsilon-technique penalty function transformation resulting in a highly nonlinear programming problem, (4) finite difference discretization of the state equations again resulting in a nonlinear program, (5) second variation methods with complex state differential equations to include damping effects resulting in systems of inhomogeneous matrix Riccatti equations some of which are stiff, (6) quasi-linear methods based on iterative linearization of the state and adjoint equation. The paper includes a discussion of some substantial computational difficulties encountered in the implementation of these techniques together with a resume of work presently in progress using a differential dynamic programming approach.
Desilva, B. M. E.↗
A Spherical Earth Solution for TOA Lightning Location Retrieval
With the advent of high-speed digital computers, the method of chi square minimization is a highly practical means for analyzing a wide variety of (otherwise intractable) nonlinear inversion problems in applied mathematical physics. Little thought or effort is required to apply the chi square method to obtain quick and reasonable estimates of a solution, and the method offers a means to assess retrieval errors. Because the method is simple and practical it is sometimes hastily applied to problems that can be solved by formal analytic or quasi-analytic means. Presently, Global Atmospherics Inc. (GAI) finds the minimum of a chi square function to analyze time-of-arrival (TOA) and magnetic bearing data derived from the National Lightning Detection Network (NLDN); ellipsoidal Earth geometry is assumed. An analytic solution to this problem has not yet been attained, but the consideration and solving of less general problem statements might eventually lead to a final solution. In the present study, the problem of retrieving lightning ground-strike location on a spherical Earth surface using a network of 4 or more time-of-arrival (TOA) sensors is considered. It is shown that this problem has an analytic solution and therefore does not require the use of nonlinear estimation theory (such as the chi square method mentioned above). The mathematical robustness of the analytic solution is tested using computer-generated lightning sources and simulated TOA measurement errors. A quasi-analytic extension of the spherical Earth solution for an oblate spheroidal Earth geometry is considered in a related study. The incorporation of magnetic bearing information into these analytic solutions would lead to a general and elegant analytic retrieval scheme that would most likely replace the chi square estimation theory currently employed by Global Atmospherics Inc. (GAI).
Koshak, William J,↗
Optimal Low Thrust Trajectories Using Differential Inclusion Concepts
A direct optimization method based on differential inclusion concepts has been developed and used to compute low thrust trajectories. This new formulation removes explicit control dependence from the problem statement thereby reducing the dimension of the parameter space of the resulting nonlinear programming problem. A simple example of a two-dimensional gravity-free trajectory involving a maximum velocity transfer to a rectilinear path is discussed. Three interplanetary trajectory examples, an Earth-Mars constant specific impulse transfer, and Earth-Jupiter constant specific impulse transfer, and an Earth-Venus-Mars variable specific impulse gravity assist, are also included. An analysis of the technique's performance is provided.
low↗
Robust Stabilization of a Class of passive Nonlinear Systems
The problem of feedback stabilization is considered for a class of nonlinear, finite dimensional, time invariant passive systems that are affine in control. Using extensions of the Kalman-Yakubovch lemma, it is shown that such systems can be stabilized by a class of finite demensional, linear, time-invariant controllers which are strictly positive real in the weak or marginal sense. The stability holds regardless of model uncertainties, and is therefore, robust.
Joshi, Suresh M.↗
Inference of Stochastic Nonlinear Oscillators with Applications to Physiological Problems
A new method of inferencing of coupled stochastic nonlinear oscillators is described. The technique does not require extensive global optimization, provides optimal compensation for noise-induced errors and is robust in a broad range of dynamical models. We illustrate the main ideas of the technique by inferencing a model of five globally and locally coupled noisy oscillators. Specific modifications of the technique for inferencing hidden degrees of freedom of coupled nonlinear oscillators is discussed in the context of physiological applications.
Smelyanskiy, Vadim N.↗
Optimality criteria solution strategies in multiple constraint design optimization
Procedures and solution strategies are described to solve the conventional structural optimization problem using the Lagrange multiplier technique. The multipliers, obtained through solution of an auxiliary nonlinear optimization problem, lead to optimality criteria to determine the design variables. It is shown that this procedure is essentially equivalent to an alternative formulation using a dual method Lagrangian function objective. Although mathematical formulations are straight-forward, successful applications and computational efficiency depend upon execution procedure strategies. Strategies examined, with application examples, include selection of active constraints, move limits, line search procedures, and side constraint boundaries.
Levy, R.↗
Nonlinear effects in the two-dimensional adaptive-wall outer-flow problem
The importance of nonlinear effects to the solution of two-dimensional adaptive-wall outer-flow problems is presently evaluated by comparing outer-flow solutions computed on the basis of the transonic small perturbation equations with solutions based on the linear Prandtl-Glauert equation. Both methods are applied to simulated measurements of transonic flow past a two-dimensional airfoil in free air. Nonlinear effects are found to be important in the outer-flow solution only where the outer flow included supersonic flow.
Schairer, Edward T.↗
Solution of transient optimization problems by using an algorithm based on nonlinear programming
An algorithm is presented for solution of dynamic optimization problems which are nonlinear in the state variables and linear in the control variables. It is shown that the optimal control is bang-bang. A nominal bang-bang solution is found which satisfies the system equations and constraints, and influence functions are generated which check the optimality of the solution. Nonlinear optimization (gradient search) techniques are used to find the optimal solution. The algorithm is used to find a minimum time acceleration for a turbofan engine.
Teren, F.↗
Assessment of shock capturing schemes for resonant flows in nonlinear instability analysis
The paper presents computational assessment of advanced numerical schemes for nonlinear acoustic problems related to combustion instabilities in liquid rocket engines. Several time-accurate, shock capturing schemes have been evaluated on a benchmark, closed-end resonant pipe flow problem. It involves the numerical solution of inviscid, compressible gas dynamics equations to predict acoustic wave propagation, wave steepening, formation of shocks, acoustic energy dissipation and wave-wall reflection for several hundred wave cycles. It was demonstrated that high accuracy TVD type schemes can be used for direct, exact nonlinear analysis of combustion instability problems, preserving high harmonic energy content for long periods of time. The selected scheme was then applied to analyze the acoustic responses of resonant pipe-resonator, radial acoustic modes and hub-baffle configurations. Interesting observations of wave shape and damping characteristics have been drawn from presented computational studies.
Przekwas, A. J.↗
A nonlinear analysis of an equilibrium craze. I - Problem formulation and solution. II - Simulations of craze and crack growth
A numerical method is developed for the determinations of the equilibrium shape of a craze in an infinite elastic plane whose fibrils exhibit very general nonlinear force-displacement behavior. The problem formulation is based on the superposition of the relevant elasticity Green's function; the solution of the resulting nonlinear problem is obtained by using Picard's successive approximation scheme. The model is used to investigate the effect of nonlinear fibril behavior on the mechanics of craze and crack growth, and the results are compared with the Dugdale model.
Ungsuwarungsri, T.↗
Energy and maximum norm estimates for nonlinear conservation laws
We have devised a technique that makes it possible to obtain energy estimates for initial-boundary value problems for nonlinear conservation laws. The two major tools to achieve the energy estimates are a certain splitting of the flux vector derivative f(u)(sub x), and a structural hypothesis, referred to as a cone condition, on the flux vector f(u). These hypotheses are fulfilled for many equations that occur in practice, such as the Euler equations of gas dynamics. It should be noted that the energy estimates are obtained without any assumptions on the gradient of the solution u. The results extend to weak solutions that are obtained as point wise limits of vanishing viscosity solutions. As a byproduct we obtain explicit expressions for the entropy function and the entropy flux of symmetrizable systems of conservation laws. Under certain circumstances the proposed technique can be applied repeatedly so as to yield estimates in the maximum norm.
Olsson, Pelle↗
Nonlinear finite elements
Comprehensive survey of past and current literature on geometrically-nonlinear finite elements is organized into handbook form and serves as valuable reference when solving problems in nonlinear structural mechanics. Handbook provides rapid access to wide variety of element types and facilitates evaluation of different elements as to their features, probable accuracy, and complexity.
Jones, R. E.↗
Algorithmic Perspectives on Problem Formulations in MDO
This work is concerned with an approach to formulating the multidisciplinary optimization (MDO) problem that reflects an algorithmic perspective on MDO problem solution. The algorithmic perspective focuses on formulating the problem in light of the abilities and inabilities of optimization algorithms, so that the resulting nonlinear programming problem can be solved reliably and efficiently by conventional optimization techniques. We propose a modular approach to formulating MDO problems that takes advantage of the problem structure, maximizes the autonomy of implementation, and allows for multiple easily interchangeable problem statements to be used depending on the available resources and the characteristics of the application problem.
Alexandrov, Natalia M.↗
On the Dynamics of Computing a Chemically Relaxed Nonequilibrium Flow
In order to gain insights into the strong dependence of numerical solutions on initial data for finite time steps, a set of nonlinear test problems rich enough to capture the behavior of difference schemes were recently identified and the numerical basins of attraction for these problems were computed using commonly used time discretizations in CFD. Our study revealed a wealth of surprisingly nonlinear behavior of numerical schemes that were not observed before, in particular for the implicit time discretizations that are commonly used in CFD. The goal of this work is to apply these tools to study a practical model from non-equilibrium flowfield relaxation. This type of problem arises in chemically nonequilibrium hypersonic flows such as in a shock tube experiment or an expanding nozzle. Here we consider a reacting mixture of (N2, N) for an inviscid one-dimensional steady model. Preliminary numerical results indicate that, aside from the possibility of spurious numerical solutions being introduced by the time discretizations, limitations on the model for physical or accurate solutions may also play a part in the dynamics observed.
Sweby, P. K.↗