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Method for first estimate of initial values of Lagrange multipliers for two point boundary value problem of calculus of variations
Lagrange multiplier matrix in minimum weight and fully stressed optimum structural design techniques
Transformation of Lagrange multipliers and Euler equations to obtain control angle solutions by variational calculus
Lagrange multiplier technique for determining stationary points /of functions/ or stationary functions /of integrals/ of expected value of random functions with certain random constraints
A coupled multiple-method analysis procedure for use late in conceptual design or early in preliminary design of aircraft structures is described. Using this method, aircraft wing structures are represented with equivalent plate models, and structural details such as engine/pylon structure, landing gear, or a 'stick' model of a fuselage are represented with beam finite element models. These two analysis methods are implemented in an integrated multiple-method formulation that involves the assembly and solution of a combined set of linear equations. The corresponding solution vector contains coefficients of the polynomials that describe the deflection of the wing and also the components of translations and rotations at the joints of the beam members. Two alternative approaches for coupling the methods are investigated; one using transition finite elements and the other using Lagrange multipliers. The coupled formulation is applied to the static analysis and vibration analysis of a conceptual design model of a fighter aircraft. The results from the coupled method are compared with corresponding results from an analysis in which the entire model is composed of finite elements.
A steepest descents optimization program is applied to the problem of a lifting vehicle entering the earth's atmosphere. The program employs penalty functions representing terminal conditions and inflight inequality constraints. During each iteration, it reduces a single performance measure which is the sum of the performance index and the penalty functions. Therefore, only one set of adjoint equations must be integrated per iteration. Values of weight factors, multiplying the penalty functions, are automatically adjusted before each iteration in order that the penalty functions will approach acceptable values. This method is shown to be a form of the classical Lagrange multiplier methods.
A method and computer program were developed for calculating the creep and optimizing the dimensions of capsules filled with alpha-emitting radioisotopes. The method solves an integral equation that was developed assuming linear accumulation of partial creep lives and relating life to time-dependent stress and temperature using the Larson-Miller parameter. The computer program, CAPSUL, is written in Fortran language for the IBM 360/75 computer. The program makes a least squares fit of the creep life function using conventional constant stress, constant temperature creep data. Dimensions of capsules having maximum thermal power per unit of weight, volume, or area are calculated for a given creep life and pressure-temperature history using a numerical Lagrange Multiplier formulation. The program also calculates the life to a prescribed strain for capsules of given dimensions and pressure-temperature history. The method has been used to analyze creep data for the alloys 304 stainless steel, Hastelloy N, Cb-1% Zr, FS-85, and T-222.
Space vehicle docking dynamics and matrix equations of motion amenable to numerical solution on digital computer, using Hamilton principle and Lagrange multipliers
Single set of mathematical expressions describes system cost and probability of error of data transmission in terms of four basic parameters in the link equation. A Lagrange multiplier sets up equations whose solutions yield the optimum values for system design considerations and weight and cost values.
Indirect methods of trajectory optimization, improving convergence through normalization of state data and Lagrange multipliers
Coordinate systems generation for optimal trajectories, considering Lagrange multiplier as motion constant
Computer program for transfer orbit trajectory optimization based on Lagrange multipliers, Newton-Raphson method, and power series
Lagrange multiplier in Banach space for settling optimal control in time lag system
An original technique for determining the optimal magnetic torque strategy for control of the attitude of spin stabilized spacecraft is presented. By employing Lagrange multipliers and the Calculus of Variations, optimal control equations are derived which define minimum time and minimum energy attitude maneuvers. Computer program algorithms to numerically solve these optimal control equations are also described. The performance of this technique is compared with a commonly employed planning method.
The equations of motion and a computer program for the dynamics of a six degree of freedom body joined to a five degree of freedom body by a quasilinear elastic tether are presented. The forebody is assumed to be a completely general rigid body with six degrees of freedom; the decelerator is also assumed to be rigid, but with only five degrees of freedom (symmetric about its longitudinal axis). The tether is represented by a spring and dashpot in parallel, where the spring constant is a function of tether elongation. Lagrange's equation is used to derive the equations of motion with the Lagrange multiplier technique used to express the constraint provided by the tether. A computer program is included which provides a time history of the dynamics of both bodies and the tension in the tether.
It is shown that it is possible to make a change of variables in a Lagrangian in such a way that the number of variables is increased. The Euler-Lagrange equations in the redundant variables are obtained in the standard way (without the use of Lagrange multipliers). These equations are not independent but they are all valid and consistent. In some cases they are simpler than if the minimum number of variables are used. The redundant variables are supposed to be related to each other by several constraints (not necessarily holonomic), but these constraints are not used in the derivation of the equations of motion. The method is illustrated with the well known Kustaanheimo-Stiefel regularization. Some interesting applications to perturbation theory are also described.
The purpose of this paper is to show how Lagrange multipliers can be used with finite elements to achieve a number of desirable properties in the underlying approximation. For elliptic boundary value problems, variational principles can be developed in which all boundary conditions are natural. In fluid flow problems, one can endow the approximations with physically essential conservation laws.
Prospective techniques for analyzing magnetic fields using NASTRAN are reviewed. A variational principle utilizing a vector potential function is presented which has as its Euler equations, the required field equations and boundary conditions for static magnetic fields including current sources. The need for an addition to this variational principle of a constraint condition is discussed. Some results using the Lagrange multiplier method to apply the constraint and CELAS2 elements to simulate the matrices are given. Practical considerations of using large numbers of CELAS2 elements are discussed.