Results for “variational principles” — Page 4

NASA NTRS1977

Variational energy principle for compressible, baroclinic flow. 1: First and second variations of total kinetic action

The case of a cold gas in the absence of external force fields is considered. Since the only energy involved is kinetic energy, the total kinetic action (i.e., the space-time integral of the kinetic energy density) should serve as the total free-energy functional in this case, and as such should be a local minimum for all possible fluctuations about stable flow. This conjecture is tested by calculating explicit, manifestly covariant expressions for the first and second variations of the total kinetic action in the context of Lagrangian kinematics. The general question of the correlation between physical stability and the convexity of any action integral that can be interpreted as the total free-energy functional of the flow is discussed and illustrated for the cases of rectillinear and rotating shearing flows.

Schmid, L. A.↗

NASA NTRS1989

A weak Hamiltonian finite element method for optimal control problems

A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.

Hodges, Dewey H.↗

NASA NTRS1991

Weak Hamiltonian finite element method for optimal control problems

A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.

Hodges, Dewey H.↗

NASA NTRS1990

A weak Hamiltonian finite element method for optimal control problems

A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.

Hodges, Dewey H.↗

NASA NTRS1993

Finite element analysis of time-independent superconductivity

The development of electromagnetic (EM) finite elements based upon a generalized four-potential variational principle is presented. The use of the four-potential variational principle allows for downstream coupling of EM fields with the thermal, mechanical, and quantum effects exhibited by superconducting materials. The use of variational methods to model an EM system allows for a greater range of applications than just the superconducting problem. The four-potential variational principle can be used to solve a broader range of EM problems than any of the currently available formulations. It also reduces the number of independent variables from six to four while easily dealing with conductor/insulator interfaces. This methodology was applied to a range of EM field problems. Results from all these problems predict EM quantities exceptionally well and are consistent with the expected physical behavior.

Schuler, James J.↗

DOE OSTI2021

Geometric electrostatic particle-in-cell algorithm on unstructured meshes

We present a geometric particle-in-cell (PIC) algorithm on unstructured meshes for studying electrostatic perturbations with frequency lower than electron gyrofrequency in magnetized plasmas. In this method, ions are treated as fully kinetic particles and electrons are described by the adiabatic response. The PIC method is derived from a discrete variational principle on unstructured meshes. To preserve the geometric structure of the system, the discrete variational principle requires that the electric field is interpolated using Whitney 1-forms, the charge is deposited using Whitney 0-forms and the electric field is computed by discrete exterior calculus. The algorithm has been applied to study the ion Bernstein wave (IBW) in two-dimensional magnetized plasmas. The simulated dispersion relations of the IBW in a rectangular region agree well with theoretical results. In a two-dimensional circular region with fixed boundary condition, the spectrum and eigenmode structures of the IBW are obtained from simulations. We compare the energy conservation property of the geometric PIC algorithm derived from the discrete variational principle with that of previous PIC methods on unstructured meshes. The comparison shows that the new PIC algorithm significantly improves the energy conservation property.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1987

Equation-of-state spinning fluids in the Einstein-Cartan theory

The relativistic fluid equations may be completed in two physically distinct methods. One method assumes the mass, rho, (or particle number) is conserved, while the other method assumes an equation of state of the form P = P(rho). A variational principle for the mass conservation method both with and without an intrinsic spin for the fluid was constructed earlier (Ray and Smalley, 1982 and 1983). A variational principle for the fluid described by an equation of state both with and without spin is formulated. In all cases the variational principle is set in the Einstein-Cartan metric-torsion U4 geometry. The results for general relativity follow as a special case.

Ray, John R.↗

NASA NTRS1982

A finite element formulation of Euler equations for the solution of steady transonic flows

The main objective of the considered investigation is related to the development of a relaxation scheme for the analysis of inviscid, rotational, transonic flow problems. To formulate the equations of motion for inviscid flows in a fixed coordinate system, an Eulerian type variational principle is required. The derivation of an Eulerian variational principle which is employed in the finite element formulation is discussed. The presented numerical method describes the mathematical formulation and the application of a numerical process for the direct solution of steady Euler equations. The development of the procedure as an extension of existing potential flow formulations provides the applicability of previous procedures, e.g., proper application of the artificial viscosity for supersonic elements, and the accurate modeling of the shock.

Ecer, A.↗

DOE OSTI2023

When action is not least for systems with action-dependent Lagrangians

The dynamics of some non-conservative and dissipative systems can be derived by calculating the first variation of an action-dependent action according to the variational principle of Herglotz. This is directly analogous to the variational principle of Hamilton commonly used to derive the dynamics of conservative systems. In a similar fashion, just as the second variation of a conservative system’s action can be used to infer whether that system’s possible trajectories are dynamically stable, so too can the second variation of the action-dependent action be used to infer whether the possible trajectories of non-conservative and dissipative systems are dynamically stable. In this paper, I show, generalizing earlier analyses of the second variation of the action for conservative systems, how to calculate the second variation of the action-dependent action and how to apply it to two physically important systems: a time-independent harmonic oscillator and a time-dependent harmonic oscillator.

Physics↗

NASA NTRS1971

Dynamic behavior of the mercury damper

The dynamic behavior of the mercury nutation damper is investigated. Particular attention is paid to the eccentric annular mercury configuration, which is the final continuous ring phase that occurs in the operation of all mercury dampers. In this phase, damping is poorest, and the system is closely linear. During the investigation, the hydrodynamic problem is treated as three dimensional, and extensive use is made of a variational principle of least-viscous frictional power loss. A variational principle of least-constraint is also used to advantage. Formulas for calculating the behavior of the mercury damper are obtained. Some confirmatory experiments were performed with transparent ring channels on a laboratory gyroscope. Selected movie frames taken during wobble damping are shown along with the results of film measurements.

Crout, P. D.↗

NASA NTRS1988

Quantum mechanical algebraic variational methods for inelastic and reactive molecular collisions

The quantum mechanical problem of reactive or nonreactive scattering of atoms and molecules is formulated in terms of square-integrable basis sets with variational expressions for the reactance matrix. Several formulations involving expansions of the wave function (the Schwinger variational principle) or amplitude density (a generalization of the Newton variational principle), single-channel or multichannel distortion potentials, and primitive or contracted basis functions are presented and tested. The test results, for inelastic and reactive atom-diatom collisions, suggest that the methods may be useful for a variety of collision calculations and may allow the accurate quantal treatment of systems for which other available methods would be prohibitively expensive.

Schwenke, David W.↗

NASA NTRS1990

Dynamical basis sets for algebraic variational calculations in quantum-mechanical scattering theory

New basis sets are proposed for linear algebraic variational calculations of transition amplitudes in quantum-mechanical scattering problems. These basis sets are hybrids of those that yield the Kohn variational principle (KVP) and those that yield the generalized Newton variational principle (GNVP) when substituted in Schlessinger's stationary expression for the T operator. Trial calculations show that efficiencies almost as great as that of the GNVP and much greater than the KVP can be obtained, even for basis sets with the majority of the members independent of energy.

Sun, Yan↗

NASA NTRS1986

Electromagnetic scattering theory

Electromagnetic scattering theory is discussed with emphasis on the general stochastic variational principle (SVP) and its applications. The stochastic version of the Schwinger-type variational principle is presented, and explicit expressions for its integrals are considered. Results are summarized for scalar wave scattering from a classic rough-surface model and for vector wave scattering from a random dielectric-body model. Also considered are the selection of trial functions and the variational improvement of the Kirchhoff short-wave approximation appropriate to large size-parameters. Other applications of vector field theory discussed include a general vision theory and the analysis of hydromagnetism induced by ocean motion across the geomagnetic field. Levitational force-torque in the magnetic suspension of the disturbance compensation system (DISCOS), now deployed in NOVA satellites, is also analyzed using the developed theory.

Bird, J. F.↗

NASA NTRS1991

A variational justification of the assumed natural strain formulation of finite elements

The objective is to study the assumed natural strain (ANS) formulation of finite elements from a variational standpoint. The study is based on two hybrid extensions of the Reissner-type functional that uses strains and displacements as independent fields. One of the forms is a genuine variational principle that contains an independent boundary traction field, whereas the other one represents a restricted variational principle. Two procedures for element level elimination of the strain field are discussed, and one of them is shown to be equivalent to the inclusion of incompatible displacement modes. Also, the 4-node C(exp 0) plate bending quadrilateral element is used to illustrate applications of this theory.

Militello, Carmelo↗

NASA NTRS1983

Spinning fluids in the Einstein-Cartan theory

An Eulerian variational principle for a spinning fluid in the Einstein-Cartan metric-torsion theory is presented. The variational principle yields the complete set of field equations for the system. The symmetric energy-momentum tensor is a sum of a perfect-fluid term and a spin term.

Ray, J. R.↗

NASA NTRS1965

Sloshing of liquids in axisymmetric ellipsoidal tanks.

Natural frequencies and modes of oscillation of liquids in partially filled axisymmetric ellipsoidal tanks calculated, using variational principle

VARIATIONAL PRINCIPLE↗

NASA NTRS2022

Computing Sensitivities in Evolutionary Systems: A Real-time Reduced Order Modeling Strategy

We present a new methodology for computing sensitivities in evolutionary systems using a model-driven low-rank approximation. To this end, we formulate a variational principle that seeks to minimize the distance between the time derivative of the reduced approximation and sensitivity dynamics. The first order optimality condition of the variational principle leads to a system of closed form evolution equations for an orthonormal basis and corresponding sensitivity coefficients. This approach allows for the computation of sensitivities with respect to a large number of parameters in an accurate and tractable manner by extracting correlations between different sensitivities on the fly. The presented method requires solving forward evolution equations, sidestepping the restrictions imposed by the forward/backward workflow of adjoint sensitivities. For example, the presented method, unlike the adjoint equation, does not impose any input/output load and can be used in applications in which real-time sensitivities are of interest. We demonstrate the utility of the method for three test cases: (1) computing sensitivity with respect to model parameters in the Rössler system, (2) computing sensitivity with respect to an infinite-dimensional forcing parameter in the chaotic Kuramoto--Sivashinsky equation, and (3) computing sensitivity with respect to reaction parameters for species transport in a turbulent reacting flow.

Reduced order model↗

NASA NTRS1993

Coupled structural, thermal, phase-change and electromagnetic analysis for superconductors, volume 2

Two families of parametrized mixed variational principles for linear electromagnetodynamics are constructed. The first family is applicable when the current density distribution is known a priori. Its six independent fields are magnetic intensity and flux density, magnetic potential, electric intensity and flux density and electric potential. Through appropriate specialization of parameters the first principle reduces to more conventional principles proposed in the literature. The second family is appropriate when the current density distribution and a conjugate Lagrange multiplier field are adjoined, giving a total of eight independently varied fields. In this case it is shown that a conventional variational principle exists only in the time-independent (static) case. Several static functionals with reduced number of varied fields are presented. The application of one of these principles to construct finite elements with current prediction capabilities is illustrated with a numerical example.

Felippa, Carlos A.↗