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Lagrangian

If the computed step size exceeds the trust radius, (, its direction is reoptunized under the condition that Aq = t, i.e., the Lagrangian... [Pg.2338]

A better approach is the method of Lagrange multipliers. This introduces the Lagrangian fiinction [59]... [Pg.2348]

Another possibility to represent the quantum mechanical Lagrangian density is using the logarithm of the amplitude X = Ina, a = e. In that particular representation, the Lagrangean density takes the following symmetrical fomi... [Pg.161]

We thus obtain a Lagrangean density, whieh is equivalent to Eq. (149) for all solutions of the Dirac equation, and has the structure of the nonrelativistic Lagrangian density, Eq. (140). Its variational derivations with respect to v / and v / lead to the solutions shown in Eq. (152), as well as to other solutions. [Pg.163]

By using Eq. (5), we can write the Lagrangian in a more symmetric fomi as... [Pg.224]

The key to these more efficient treatments is a natural canonical formulation of the rigid body dynamics in terms of rotation matrices. The orientational term of the Lagrangian in these variables can be written simply as... [Pg.352]

We now consider the formulation of the equations of motion for a rigid body pinned at its center of mass and acted on by a (possibly nonlinear) potential field. The Lagrangian in this case is... [Pg.354]

A Hamiltonian version of the quaternionic description is also possible by viewing the quaternions as a set of generalized coordinates, introducing those variables into the rigid body Lagrangian (1), and finally determining the canonical momenta through the formula... [Pg.355]

Rotation matrices may be viewed as an alternative to particles. This approach is based directly on the orientational Lagrangian (1). Viewing the elements of the rotation matrix as the coordinates of the body, we directly enforce the constraint Q Q = E. Introducing the canonical momenta P in the usual manner, there results a constrained Hamiltonian formulation which is again treatable by SHAKE/RATTLE [25, 27, 20]. For a single rigid body we arrive at equations for the orientation of the form[25, 27]... [Pg.356]

In order to minimise the energy we introduce this constraint as a Lagrangian multiplier I /I), leading to ... [Pg.147]

The constraint force can be introduced into Newton s equations as a Lagrange multipli (see Section 1.10.5). To achieve consistency with the usual Lagrangian notation, we wri F y as —A and so F Ar equals Am. Thus ... [Pg.387]

Using two-noded Lagrangian elements the shape functions are given as... [Pg.59]

As mentioned in Chapter 1, in general, the solution of the integral viscoelastic models should be based on Lagrangian frameworks. In certain types of flow... [Pg.86]

The geometrical flexibility of the VOF scheme can be significantly improved if in its formulation, instead of using a fixed framework, a combination of a Lagrangian-Eulerian approach is adopted. The most common approach to develop such a combined framework is the application of the Arbitrary... [Pg.102]

Lagrangian-Eulerian (ALE) method. In the ALE technique the finite element mesh used in the simulation is moved, in each time step, according to a predetermined pattern. In this procedure the element and node numbers and nodal connectivity remain constant but the shape and/or position of the elements change from one time step to the next. Therefore the solution mesh appears to move with a velocity which is different from the flow velocity. Components of the mesh velocity are time derivatives of nodal coordinate displacements expressed in a two-dimensional Cartesian system as... [Pg.103]

FINITE ELEMENT MODELLENG OF POLYMERIC FLOW PROCESSES 3.5.3 VOF method in Lagrangian frameworks... [Pg.104]

Donea, J., 1992. Arbitrary Lagrangian-Eulerian finite element methods. In Belytschko, T. and Hughes, T. J. R. (eds), Computational Methods for Transient Analysis, Elsevier Science, Amsterdam. [Pg.108]

Figure 5.4 The finite element mesh configurations in the Arbitrary Lagrangian-Eulerian scheme... Figure 5.4 The finite element mesh configurations in the Arbitrary Lagrangian-Eulerian scheme...
Note that in a Lagrangian system the convection terms in Equation (5.11) vanish. [Pg.153]

Note that the shape functions used in the above discretization preserv c their originally defined forms. This i.s in contrast to the Lagrangian formulations in which the shape functions need to be modified (Donea and Qiuirtapellc, 1992). [Pg.154]

The relationship between H and vibrational frequencies can be made clear by recalling the classical equations of motion in the Lagrangian formulation ... [Pg.516]

Applying the Lagrangian equations to this form for L gives the equations of motion of the qj coordinates ... [Pg.516]

A similar effect is obtained by using the spin-constrained UHF method (SUHF). In this method, the spin contamination error in a UHF wave function is constrained by the use of a Lagrangian multiplier. This removes the spin contamination completely as the multiplier goes to infinity. In practice, small positive values remove most of the spin contamination. [Pg.229]


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A Lagrangian Box Model

Acceleration Lagrangian

Action integrals Lagrangian

Arbitrary Lagrangian-Eulerian

Arbitrary-Lagrangian-Eulerian (ALE) Codes

Atomic action and Lagrangian integrals

Augmented Lagrangian Penalty Function

Augmented Lagrangian methods

Biodynamics a Lagrangian Approach

Bound-Constrained Formulation for Lagrangian Penalty Function

Car-Parrinello lagrangian

Classical Lagrangian

Classical Lagrangian and Hamiltonian

Classical mechanics Lagrangian formalism

Classical mechanics lagrangian

Comparison of Eulerian and Lagrangian Approaches

Control volume Lagrangian

Coordinate system Lagrangian

Correlation functions Lagrangian

Coupled cluster Lagrangian

Definition of the Lagrangian density

Euler-Lagrangian

Eulerian and Lagrangian Coordinates

Eulerian computation, compared with Lagrangian

Eulerian grid to Lagrangian positions

Eulerian-Lagrangian approach

Eulerian-Lagrangian approach fraction

Eulerian-Lagrangian approach simulation example

Eulerian-Lagrangian approach time steps

Eulerian-Lagrangian approach transfer

Eulerian-Lagrangian framework

Eulerian-Lagrangian methods

Eulerian-Lagrangian methods multiphase flows

Eulerian-Lagrangian model

Extended Lagrangian

Extended Lagrangian method

Extended Lagrangian technique

Extended Lagrangians

Extended-Lagrangian formalism

Fast Lagrangian Analysis of Continua

Fermi Lagrangian

Formulation, semi-Lagrangian

Framework Lagrangian

Gauge Transformations of the Lagrangian

Gauge field fermion Lagrangian

Generalized Lagrangian strain measure

Grid, Lagrangian

Induced dipoles extended Lagrangian method

Interaction Lagrangian

Inverse Lagrangian Analysis of Isotopic Composition

LaGrangian flow

LaGrangian particle tracking

Lagrange, Lagrangian

Lagrangian Autocorrelations of Fluctuating Velocities

Lagrangian Equation of Equilibrium

Lagrangian Oscillator Dynamics

Lagrangian PDF methods

Lagrangian Techniques

Lagrangian Trajectory Approach

Lagrangian and Hamiltonian Formulation

Lagrangian and Hamiltonian Mechanics

Lagrangian approach

Lagrangian approach, turbulent diffusion

Lagrangian augmented function

Lagrangian bound-constrained

Lagrangian chaos

Lagrangian classical physics

Lagrangian computation, compared

Lagrangian computer code

Lagrangian conservation relations

Lagrangian coordinates

Lagrangian correlation coefficient

Lagrangian correspondence

Lagrangian covariant

Lagrangian decomposition

Lagrangian density

Lagrangian density interactions

Lagrangian density matrix

Lagrangian density mechanics

Lagrangian density molecular systems

Lagrangian derivative

Lagrangian description

Lagrangian dual problem

Lagrangian dynamics

Lagrangian dynamics approach

Lagrangian electromagnetic field

Lagrangian element

Lagrangian equation 0 electrodynamics

Lagrangian equation classical mechanics

Lagrangian equation corrections

Lagrangian equation equilibrium

Lagrangian equation limits

Lagrangian equation of motion

Lagrangian equation quantum mechanics

Lagrangian equations

Lagrangian equations action integrals

Lagrangian equations connection

Lagrangian equations density

Lagrangian equations dynamics

Lagrangian equations motion

Lagrangian equations path integrals

Lagrangian equations quantum dynamics

Lagrangian equations theory

Lagrangian equations time-dependent expressions

Lagrangian equations value formulation

Lagrangian equations, symmetric states

Lagrangian field theory

Lagrangian fluctuating velocity

Lagrangian for Electromagnetic Interactions

Lagrangian for the Electrodynamic Field

Lagrangian form

Lagrangian formalism

Lagrangian formalism of polymer theory

Lagrangian formalism of the field theory

Lagrangian formulation

Lagrangian frame

Lagrangian frame of reference

Lagrangian function

Lagrangian function classical mechanics

Lagrangian function optimization

Lagrangian function, definition

Lagrangian functional

Lagrangian gauge invariance

Lagrangian gauge transformation

Lagrangian integral time scale

Lagrangian linear programming

Lagrangian linear systems

Lagrangian local parameter

Lagrangian manifold

Lagrangian marker particles

Lagrangian measurements

Lagrangian mechanics

Lagrangian methods

Lagrangian micromixing models

Lagrangian microscale

Lagrangian minimization

Lagrangian mode

Lagrangian models for the micromixing rate

Lagrangian motion

Lagrangian multiplier

Lagrangian multiplier expansion

Lagrangian multiplier method

Lagrangian numerical scheme

Lagrangian particle model

Lagrangian penalty function

Lagrangian plane

Lagrangian points

Lagrangian properties

Lagrangian properties defined

Lagrangian properties particle models

Lagrangian reference frame

Lagrangian relativistic

Lagrangian relaxation

Lagrangian representation

Lagrangian sectional approach

Lagrangian single-particle model

Lagrangian solution

Lagrangian spin

Lagrangian strain

Lagrangian strain measure

Lagrangian strong form

Lagrangian theory

Lagrangian time-dependent

Lagrangian time-distance

Lagrangian time-distance diagram

Lagrangian trace

Lagrangian trajectory model

Lagrangian transform

Lagrangian transport

Lagrangian triads

Lagrangian turbulence

Lagrangian variational

Lagrangian vector multiplier

Lagrangian velocity autocorrelation

Lagrangian viewpoint

Lagrangian weak form

Lagrangian, nonlinear

Lagrangians

Lagrangians

Lagrangians coupled-cluster methods

Mechanics of Constrained Systems within Lagrangian and Hamiltonian Formalisms

Method of Lagrangian multipliers

Mixed Eulerian-Lagrangian method

Model arbitrary Lagrangian Eulerian method

Models LaGrangian particle tracking

Models Lagrangian

Models Lagrangian Photochemical

Molecular dynamics extended Lagrangian methods

Molecular dynamics simulations extended Lagrangian method

One-Dimensional Lagrangian Models

Optimization Lagrangian

Optimization augmented Lagrangian method

Particle size distribution Lagrangian approach

Particle tracking Lagrangian approach

Perturbation theory Lagrangian

Properties of the quantum mechanical Lagrangian

Quasi-energy Lagrangian

Random field Lagrangian

Regional Lagrangian Model of Air Pollution

Regional Lagrangian Model of Air Pollution RELMAP)

Relativistic Lagrangian theories

Semi-Lagrangian

Semi-Lagrangian advection

Single-Point Lagrangian Equations

Stochastic process Lagrangian model

Symmetry broken electroweak Lagrangian

The CCPT Lagrangian

The Lagrangian Model

The Lagrangian and Hamiltonian

The Lagrangian density

The Lagrangian density for diffusion

The convection term. Lagrangian and Eulerian calculations

The coupled-cluster Lagrangian

The method of Lagrangian multipliers

Total Lagrangian description

Total Lagrangian form

Transformation from Lagrangian to Hamiltonian

Updated Lagrangian form

VOF method in Arbitrary Lagrangian-Eulerian frameworks

VOF method in Lagrangian frameworks

Variational Lagrangian general formulation

Velocity field Lagrangian

Virtual-mass force Lagrangian

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