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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.

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NASA NTRS1993

Domain decomposition methods for nonconforming finite element spaces of Lagrange-type

In this article, we consider the application of three popular domain decomposition methods to Lagrange-type nonconforming finite element discretizations of scalar, self-adjoint, second order elliptic equations. The additive Schwarz method of Dryja and Widlund, the vertex space method of Smith, and the balancing method of Mandel applied to nonconforming elements are shown to converge at a rate no worse than their applications to the standard conforming piecewise linear Galerkin discretization. Essentially, the theory for the nonconforming elements is inherited from the existing theory for the conforming elements with only modest modification by constructing an isomorphism between the nonconforming finite element space and a space of continuous piecewise linear functions.

Cowsar, Lawrence C.↗
NASA NTRS2020

Model-Reference Adaptive Control of Distributed Lagrangian Infinite-Dimensional Systems Using Hamilton’s Principle

This paper presents a Hamilton's principle for distributed control of infinite-dimensional systems modeled by a distributed form of the Euler-Lagrange method. The distributed systems are governed by a system of linear partial differential equations in space and time. A generalized potential energy expression is developed that can capture most physical systems including those systems that have no spatial distribution. The Hamilton's principle is applied to derive distributed feedback control methods without resorting to the standard weak-form discretization approach to convert an infinite-dimensional systems to a finite-dimensional systems. It can be shown by the principle of least action that the distributed control synthesized by the Hamilton's principle is a minimum-norm control. A model-reference adaptive control framework is developed for distributed Lagrangian systems in the presence of uncertainty. The theory is demonstrated by an application of adaptive flutter suppression control of a flexible aircraft wing.

Nguyen, Nhan T.↗
NASA NTRS1976

Numerical determination of the transmissibility characteristics of a squeeze film damped forced vibration system

Numerical solutions of the governing equations of motion of a liquid squeeze film damped forced vibration system were carried out to examine the feasibility of using a liquid squeeze film to cushion and protect large structures, such as buildings, located in areas of high seismic activity. The mathematical model used was that for a single degree of freedom squeeze film damped spring mass system. The input disturbance was simulated by curve fitting actual seismic data with an eleventh order Lagranging polynomial technique. Only the normal component of the seismic input was considered. The nonlinear, nonhomogeneous governing differential equation of motion was solved numerically to determine the transmissibility over a wide range of physical parameters using a fourth-order Runge-Kutta technique. It is determined that a liquid squeeze film used as a damping agent in a spring-mass system can significantly reduce the response amplitude for a seismic input disturbance.

Sutton, M. A.↗
NASA NTRS1994

Finite element approximation of an optimal control problem for the von Karman equations

This paper is concerned with optimal control problems for the von Karman equations with distributed controls. We first show that optimal solutions exist. We then show that Lagrange multipliers may be used to enforce the constraints and derive an optimality system from which optimal states and controls may be deduced. Finally we define finite element approximations of solutions for the optimality system and derive error estimates for the approximations.

Hou, L. Steven↗
NASA NTRS1998

A Posteriori Bounds for Linear-Functional Outputs of Crouzeix-Raviart Finite Element Discretizations of the Incompressible Stokes Problem

We present a finite element technique for the efficient generation of lower and upper bounds to outputs which are linear functionals of the solutions to the incompressible Stokes equations in two space dimensions; the finite element discretization is effected by Crouzeix-Raviart elements, the discontinuous pressure approximation of which is central to our approach. The bounds are based upon the construction of an augmented Lagrangian: the objective is a quadratic "energy" reformulation of the desired output; the constraints are the finite element equilibrium equations (including the incompressibility constraint), and the intersubdomain continuity conditions on velocity. Appeal to the dual max-min problem for appropriately chosen candidate Lagrange multipliers then yields inexpensive bounds for the output associated with a fine-mesh discretization; the Lagrange multipliers are generated by exploiting an associated coarse-mesh approximation. In addition to the requisite coarse-mesh calculations, the bound technique requires solution only of local subdomain Stokes problems on the fine-mesh. The method is illustrated for the Stokes equations, in which the outputs of interest are the flowrate past, and the lift force on, a body immersed in a channel.

Patera, Anthony T.↗
NASA NTRS1984

Discrimination of coherent features in turbulent boundary layers by the entropy method

Entropy in information theory is defined as the expected or mean value of the measure of the amount of self-information contained in the ith point of a distribution series x sub i, based on its probability of occurrence p(x sub i). If p(x sub i) is the probability of the ith state of the system in probability space, then the entropy, E(X) = - sigma p(x sub i) logp (x sub i), is a measure of the disorder in the system. Based on this concept, a method was devised which sought to minimize the entropy in a time series in order to construct the signature of the most coherent motions. The constrained minimization was performed using a Lagrange multiplier approach which resulted in the solution of a simultaneous set of non-linear coupled equations to obtain the coherent time series. The application of the method to space-time data taken by a rake of sensors in the near-wall region of a turbulent boundary layer was presented. The results yielded coherent velocity motions made up of locally decelerated or accelerated fluid having a streamwise scale of approximately 100 nu/u(tau), which is in qualitative agreement with the results from other less objective discrimination methods.

Corke, T. C.↗
NASA NTRS2007

Optimal Electrodynamic Tether Phasing Maneuvers

We study the minimum-time orbit phasing maneuver problem for a constant-current electrodynamic tether (EDT). The EDT is assumed to be a point mass and the electromagnetic forces acting on the tether are always perpendicular to the local magnetic field. After deriving and non-dimensionalizing the equations of motion, the only input parameters become current and the phase angle. Solution examples, including initial Lagrange costates, time of flight, thrust plots, and thrust angle profiles, are given for a wide range of current magnitudes and phase angles. The two-dimensional cases presented use a non-tilted magnetic dipole model, and the solutions are compared to existing literature. We are able to compare similar trajectories for a constant thrust phasing maneuver and we find that the time of flight is longer for the constant thrust case with similar initial thrust values and phase angles. Full three-dimensional solutions, which use a titled magnetic dipole model, are also analyzed for orbits with small inclinations.

Bitzer, Matthew S.↗
NASA NTRS1993

On a Lagrange-Hamilton formalism describing position and momentum uncertainties

According to Heisenberg's uncertainty relation, in quantum mechanics it is not possible to determine, simultaneously, exact values for the position and the momentum of a material system. Calculating the mean value of the Hamiltonian operator with the aid of exact analytic Gaussian wave packet solutions, these uncertainties cause an energy contribution additional to the classical energy of the system. For the harmonic oscillator, e.g., this nonclassical energy represents the ground state energy. It will be shown that this additional energy contribution can be considered as a Hamiltonian function, if it is written in appropriate variables. With the help of the usual Lagrange-Hamilton formalism known from classical particle mechanics, but now considering this new Hamiltonian function, it is possible to obtain the equations of motion for position and momentum uncertainties.

Schuch, Dieter↗
NASA NTRS1979

The shape of a pendent liquid drop

Formal solutions of the static equilibrium equations for the form of the outer surface of a pendent liquid drop are studied. An approach is adopted in which only the one-parameter family determined by vertex height (u sub 0) need be described. Attention is restricted to rotationally symmetric configurations, and all symmetric solutions are characterized for the case where the Lagrange parameter lambda is equal to zero. It is shown that for any u sub 0 the function u(r; u sub 0) can be extended as a parametric solution of a system of equations for all arc lengths, yielding a curve without limit sets or double points, and that the resulting capillary rotation surface spreads out indefinitely away from the axis r = 0. The asymptotic form of the surface in the case of large absolute values of u sub 0 is characterized quantitatively, along with the global structure of all such surfaces.

Concus, P.↗
NASA NTRS1995

Airfoil Design and Optimization by the One-Shot Method

An efficient numerical approach for the design of optimal aerodynamic shapes is presented in this paper. The objective of any optimization problem is to find the optimum of a cost function subject to a certain state equation (governing equation of the flow field) and certain side constraints. As in classical optimal control methods, the present approach introduces a costate variable (Lagrange multiplier) to evaluate the gradient of the cost function. High efficiency in reaching the optimum solution is achieved by using a multigrid technique and updating the shape in a hierarchical manner such that smooth (low-frequency) changes are done separately from high-frequency changes. Thus, the design variables are changed on a grid where their changes produce nonsmooth (high-frequency) perturbations that can be damped efficiently by the multigrid. The cost of solving the optimization problem is approximately two to three times the cost of the equivalent analysis problem.

Kuruvila, G.↗
NASA NTRS1985

Development and application of a unified balancing approach with multiple constraints

The development of a general analytic approach to constrained balancing that is consistent with past influence coefficient methods is described. The approach uses Lagrange multipliers to impose orbit and/or weight constraints; these constraints are combined with the least squares minimization process to provide a set of coupled equations that result in a single solution form for determining correction weights. Proper selection of constraints results in the capability to: (1) balance higher speeds without disturbing previously balanced modes, thru the use of modal trial weight sets; (2) balance off-critical speeds; and (3) balance decoupled modes by use of a single balance plane. If no constraints are imposed, this solution form reduces to the general weighted least squares influence coefficient method. A test facility used to examine the use of the general constrained balancing procedure and application of modal trial weight ratios is also described.

Zorzi, E. S.↗
NASA NTRS2004

Further Improvement in 3DGRAPE

3DGRAPE/AL:V2 denotes version 2 of the Three-Dimensional Grids About Anything by Poisson's Equation with Upgrades from Ames and Langley computer program. The preceding version, 3DGRAPE/AL, was described in Improved 3DGRAPE (ARC-14069) NASA Tech Briefs, Vol. 21, No. 5 (May 1997), page 66. These programs are so named because they generate volume grids by iteratively solving Poisson's Equation in three dimensions. The grids generated by the various versions of 3DGRAPE have been used in computational fluid dynamics (CFD). The main novel feature of 3DGRAPE/AL:V2 is the incorporation of an optional scheme in which anisotropic Lagrange-based trans-finite interpolation (ALBTFI) is coupled with exponential decay functions to compute and blend interior source terms. In the input to 3DGRAPE/AL:V2 the user can specify whether or not to invoke ALBTFI in combination with exponential-decay controls, angles, and cell size for controlling the character of grid lines. Of the known programs that solve elliptic partial differential equations for generating grids, 3DGRAPE/AL:V2 is the only code that offers a combination of speed and versatility with most options for controlling the densities and other characteristics of grids for CFD.

Alter, Stephen↗
NASA NTRS1989

Mixed formulation for frictionless contact problems

Simple mixed finite element models and a computational precedure are presented for the solution of frictionless contact problems. The analytical formulation is based on a form of Reissner's large rotation theory of the structure with the effects of transverse shear deformation included. The contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with the fundamental unknowns consisting of the internal forces (stress resultants), the generalized displacements, and the Lagrange multipliers associated with the contact conditions. The element characteristic array are obtained by using a modified form of the two-field Hellinger-Reissner mixed variational principle. The internal forces and the Lagrange multipliers are allowed to be discontinuous at interelement boundaries. The Newton-Raphson iterative scheme is used for the solution of the nonlinear algebraic equations, and the determination of the contact area and the contact pressures.

Noor, Ahmed K.↗
NASA NTRS1986

Higher-order flux difference splitting schemes for the Euler equations using upstream interpolations

A class of explicit two time-level, 2p + 1 space-point, (2p 1)-th order, upwind-biased flux difference splitting schemes are proposed for the numerical advection based on Lagrange's interpolation, and the method is an accord with the physical domain of dependence. A normalized Jacobian coefficient matrix is introduced to convert the schemes to hyperbolic systems of conservation laws, and approaches to make the higher-order schemes total variation stable are discussed. Accuracy and stability of the present schemes are examined, and implicit total variation diminishing schemes are developed for steady-state calculations.Application to gasdynamic problems for both steady and unsteady flows covering a wide range of Mach numbers is considered, and results for a blast wave passing a cylinder, and head-on collision of two blast waves over a circular arc, are presented. The flow patterns were found to be symmetric, and good resolution of flow structures was obtained.

Yang, J. Y.↗
NASA NTRS1984

On the computer analysis of structures and mechanical systems

The governing equations for the analysis of open branch-chain mechanical systems are developed in a form suitable for implementation in a general purpose finite element computer program. Lagrange's form of d'Alembert's principle is used to derive the system mass matrix and force vector. The generalized coordinates are selected as the unconstrained relative degrees of freedom giving the position and orientation of each slave link with respect to their master link. Each slave link may have from zero to six degrees of freedom relative to the reference frames of its master link. A strategy for automatic generation of the system mass matrix and force vector is described.

Bennett, B. E.↗
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 NTRS1983

Lagrange thermodynamic potential and intrinsic variables for He-3 He-4 dilute solutions

For a two-fluid model of dilute solutions of He-3 in liquid He-4, a thermodynamic potential is constructed that provides a Lagrangian for deriving equations of motion by a variational procedure. This Lagrangian is defined for uniform velocity fields as a (negative) Legendre transform of total internal energy, and its primary independent variables, together with their thermodynamic conjugates, are identified. Here, similarities between relations in classical physics and quantum statistical mechanics serve as a guide for developing an alternate expression for this function that reveals its character as the difference between apparent kinetic energy and intrinsic internal energy. When the He-3 concentration in the mixtures tends to zero, this expression reduces to Zilsel's formula for the Lagrangian for pure liquid He-4. An investigation of properties of the intrinsic internal energy leads to the introduction of intrinsic chemical potentials along with other intrinsic variables for the mixtures. Explicit formulas for these variables are derived for a noninteracting elementary excitation model of the fluid. Using these formulas and others also derived from quantum statistical mechanics, another equivalent expression for the Lagrangian is generated.

Jackson, H. W.↗