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All sourcesNASA NTRSDOE OSTI
NASA NTRS1946

The Lagrangian multiplier method of finding upper and lower limits to critical stresses of clamped plates

The theory of Lagrangian multipliers is applied to the problem of finding both upper and lower limits to the true compressive buckling stress of a clamped rectangular plate. The upper and lower limits thus bracket the true stress, which cannot be exactly found by the differential-equation approach. The procedure for obtaining the upper limit, which is believed to be new, presents certain advantages over the classical Rayleigh-Ritz method of finding upper limits. The theory of the lower-limit procedure has been given by Trefftz, but, in the present application, the method differs from that of Trefftz in a way that makes it inherently more quickly convergent. It is expected that in other buckling problems and in some vibration problems the Lagrangian multiplier method of finding upper and lower limits may be advantageously applied to the calculation of buckling stresses and natural frequencies.

Budiansky, Bernard↗
NASA NTRS1946

The Lagrangian Multiplier Method of Finding Upper and Lower Limits to Critical Stresses of Clamped Plates

The theory of Lagrangian multipliers is applied to the problem of finding both upper and lower limits to the true compressive buckling stress of a clamped rectangular plate. The upper and lower limits thus bracket the truss, which cannot be exactly found by the differential-equation approach. The procedure for obtaining the upper limit, which is believed to be new, presents certain advantages over the classical Raleigh-Rite method of finding upper limits. The theory of the lower-limit procedure has been given by Trefftz but, in the present application, the method differs from that of Trefftz in a way that makes it inherently more quickly convergent. It is expected that in other buckling problems and in some vibration problems problems the Lagrangian multiplier method finding upper and lower limits may be advantageously applied to the calculation of buckling stresses and natural frequencies.

Budiansky, Bernard↗
NASA NTRS1976

Trajectory optimization for the Atlas/Centaur launch vehicle

A state-of-art survey of computational techniques employed in design and optimization of trajectories for the Atlas/Centaur launch vehicle is presented. Attention is focused on the constrained optimization technique, related to Hestenes' (1969) method of multipliers, with various formulas used in updating the multipliers. Advantages of applying multiplier method to trajectory optimization, with gains in computing speed, are argued, and optimization of the HEAO-A (high energy astronomical observatory) is discussed. Open-loop atmospheric guidance strategy and closed-loop exoatmospheric pitch and yaw guidance equations are dealt with, and the full range of constraints to be observed during the flight is discussed.

Brusch, R. G.↗
NASA NTRS1973

Computer design of antenna reflectors.

Performance of paraboloidal antenna reflectors is adversely influenced by surface distortions from a perfect paraboloid, which cause pathlength variations of the RF energy beam. The structural design objective for the surface backup structure is to minimize the rms pathlength deviations for gravity loading. Two design approaches are illustrated and applied to sample antenna structure designs. These are a sectional search method and a virtual work/Lagrange multiplier method. Both are shown to provide useful performance improvements. The second, however, appears to be more suited for application to design of large antenna structures.

Levy, R.↗
NASA NTRS1996

Convergence of a Substructuring Method with LaGrange Multipliers

We analyze the convergence of a substructuring iterative method with Lagrange multipliers, proposed recently by Farhat and Roux. The method decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains and a coarse problem for the subdomain nullspace components. For linear conforming elements and preconditioning by the Dirichlet problems on the subdomains, we prove the asymptotic bound on the condition number C(1 + log(H/h))(sup gamma), gamma = 2 or 3, where h is the characteristic element size and H is the subdomain size.

Mandel, Jan↗
NASA NTRS1984

Application of augmented-Lagrangian methods in meteorology: Comparison of different conjugate-gradient codes for large-scale minimization

A Lagrange multiplier method using techniques developed by Bertsekas (1982) was applied to solving the problem of enforcing simultaneous conservation of the nonlinear integral invariants of the shallow water equations on a limited area domain. This application of nonlinear constrained optimization is of the large dimensional type and the conjugate gradient method was found to be the only computationally viable method for the unconstrained minimization. Several conjugate-gradient codes were tested and compared for increasing accuracy requirements. Robustness and computational efficiency were the principal criteria.

Navon, I. M.↗
NASA NTRS1994

Eliminating Computational Instability In Multibody Simulations

TWOBODY implements improved version of Lagrange multiplier method. Program ultilizes programming technique eliminating computational instability in multibody simulations in which Lagrange multipliers used. In technique, one uses constraint equations, instead of integration, to determine coordinates that are not independent. To illustrate technique, it includes simple mathematical model of solid rocket booster and parachute connected by frictionless swivel. Written in FORTRAN 77.

Watts, Gaines L.↗
NASA NTRS1996

Optimum Suction Distribution for Transition Control

The optimum suction distribution which gives the longest laminar region for a given total suction is computed. The goal here is to provide the designer with a method to find the best suction distribution subject to some overall constraint applied to the suction. We formulate the problem using the Lagrangian multiplier method with constraints. The resulting non-linear system of equations is solved using the Newton-Raphson technique. The computations are performed for a Blasius boundary layer on a flat-plate and crossflow cases. For the Blasius boundary layer, the optimum suction distribution peaks upstream of the maximum growth rate region and remains flat in the middle before it decreases to zero at the end of the transition point. For the stationary and travelling crossflow instability, the optimum suction peaks upstream of the maximum growth rate region and decreases gradually to zero.

Balakumar, P.↗
NASA NTRS1993

On solving the irregularly-shaped plate buckling problem

This paper develops methods for finding critical buckling loads of isotropic plates. In particular, this paper is motivated by the lack of tabular data for non-rectangular plates such as general trapezoids. High order displacement functions are used, and boundary conditions are imposed by utilizing the Lagrange multiplier method, which allows for a variety of constraint conditions. The element is derived with the use of minimum potential energy and subparametric mappings. An example is given, and several test cases are presented to show the validity of the approach in the one element case.

Smith, James P.↗
NASA NTRS2012

Nonlinear Analysis of Bonded Composite Tubular Lap Joints

The present study describes a semi-analytical solution method for predicting the geometrically nonlinear response of a bonded composite tubular single-lap joint subjected to general loading conditions. The transverse shear and normal stresses in the adhesive as well as membrane stress resultants and bending moments in the adherends are determined using this method. The method utilizes the principle of virtual work in conjunction with nonlinear thin-shell theory to model the adherends and a cylindrical shear lag model to represent the kinematics of the thin adhesive layer between the adherends. The kinematic boundary conditions are imposed by employing the Lagrange multiplier method. In the solution procedure, the displacement components for the tubular joint are approximated in terms of non-periodic and periodic B-Spline functions in the longitudinal and circumferential directions, respectively. The approach presented herein represents a rapid-solution alternative to the finite element method. The solution method was validated by comparison against a previously considered tubular single-lap joint. The steep variation of both peeling and shearing stresses near the adhesive edges was successfully captured. The applicability of the present method was also demonstrated by considering tubular bonded lap-joints subjected to pure bending and torsion.

E. Oterkus↗
NASA NTRS1976

Formulation of a methodology for power circuit design optimization

A methodology for optimizing power-processor designs is described which achieves optimization with respect to some power-processor characteristic deemed particularly desirable by the designer, such as weight or efficiency. Optimization theory based on Lagrange multipliers is reviewed together with nonlinear programming techniques employing penalty functions. The methodology, the task of which is to minimize an objective function subject to design constraints, is demonstrated with the aid of four examples: optimum-weight core selection for an inductor with a predetermined winding size, optimum-weight inductor design with a given loss constraint, optimum-loss inductor design with a given weight constraint, and a comparison of optimum-weight single- and two-stage input-filter designs with identical loss and other requirement constraints. Closed-form solutions for the first three examples are obtained by applying the Lagrange-multiplier method, but solutions for the last example are found numerically through the use of the sequential unconstrained minimization technique.

Yu, Y.↗
NASA NTRS1977

Analysis of magnetic fields using variational principles and CELAS2 elements

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.

Frye, J. W.↗
NASA NTRS1981

Power system design optimization using Lagrange multiplier techniques

An optimization technique using the Lagrange Multiplier Method is proposed to facilitate design of switching power converter systems. The essence of the optimization is to identify the optimal battery voltage level and switching frequency along with the detailed converter design so that the total system weight including the battery and the packaged converter is minimized, and concurrently all specified power circuit performances are satisfied.

Yu, Y.↗
NASA NTRS1983

Buckling and vibration of any prismatic assembly of shear and compression loaded anisotropic plates with an arbitrary supporting structure

The computer program designated 'VIPASA', which accurately treats buckling and vibration in prismatic plate assemblies with a response that varies sinusoidally in the longitudinal direction, has been found to be limited by the production of an in-plane shear loading of component plates that produces skewed mode shapes. These do not conform to desired support conditions. This problem is presently overcome through a coupling of the VIPASA stiffness matrices for different wavelength responses by means of the Lagrangian Multipliers method. The theory extends to supports at arbitrary locations, and even to the support provided by any elastic structure. The generality and capabilities of VIPASA have been retained in the computer program designated 'VICON', which permits constraints and a supporting structure consisting of any number of transverse beam columns.

Anderson, M. S.↗
NASA NTRS1984

ADS: A FORTRAN program for automated design synthesis, version 1.00

A new general-purpose optimization program for engineering design is described. ADS-1 (Automated Design Synthesis - Version 1) is a FORTRAN program for solution of nonlinear constrained optimization problems. The program is segmented into three levels, being strategy, optimizer, and one-dimensional search. At each level, several options are available so that a total of over 100 possible combinations can be created. Examples of available strategies are sequential unconstrained minimization, the Augmented Lagrange Multiplier method, and Sequential Linear Programming. Available optimizers include variable metric methods and the Method of Feasible Directions as examples and one-dimensional search options include polynomial interpolation and the Golden Section method as examples. Emphasis is placed on ease of use of the program. All information is transferred via a single parameter list. Default values are provided for all internal program parameters such as convergence criteria, and the user is given a simple means to over-ride these, if desired. The program is demonstrated with a simple structural design example.

Vanderplaats, G. N.↗