Extended Lagrange methods

Constant pressure MD can be implemented either by extended system methods that couple the dynamical system to an external variable V which is the volume of the simulation box (Andersen, 1980) or by constraint methods that use a Lagrange multiplier determined from Gauss principle (Evans and Morriss,... 

The Lagrange method can readily be extended to several constraints. 

The use of this theory in studies of nonlinear oscillations was suggested in 1929 (by Andronov). At a later date (1937) Krylov and Bogoliubov (K.B.) simplified somewhat the method of attack by a device resembling Lagrange s method of the variation of parameters, and in this form the method became useful for solving practical problems. Most of these early applications were to autonomous systems (mainly the self-excited oscillations), but later the method was extended to... 

In the application of this method to a Rankine cycle cogeneration system, generalized costing equations for the major components have been developed. Also, the utility of the method was extended by relaxing the rule that each state variable (and hence each Lagrange constraint) must correspond to an available-energy flow. The applicability was further extended by the introduction of numerical techniques necessary for the purpose of evaluating partial derivatives of steam table data. 

Attempts to extend RANS formulation to LES of two-phase combustion may be found in [318 354 317 255 292]. They are all based on a Euler-Lagrange (EL) description of the dispersed phase in which the flow is solved using an Eulerian method and the particles are tracked with a Lagrangian approach. An alternative is the Euler-Euler (EE) description, also called two-fluid approach, in which both the gas and the dispersed phases are... 

In cases where equations of motion are desired for deformable bodies, methods such as the extended Hamilton s principle may be employed. The energy is written for the system and, in addition to the terms used in Lagrange s equation, strain energy would be included. Application of Hamilton s principle will yield a set of equations of motion in the form of partial differential equations as well as the corresponding boundary conditions. Derivations and examples can be found in other sources (Baruh, 1999 Benaroya, 1998). Hamilton s principle employs the calculus of variations, and there are many texts that will be of benefit (Lanczos, 1970). 

The same authors also presented an example of the use of the population balance equation (PBE) (distribution of biomass m) coimected to the multi-zone/CFD model. This example is in several respects relevant for the assessment of the modeling approach. The coupling of the integro-differential equation of the population balance is a numerical challenge, which can nowadays be tackled within the environment of a CFD approach, albeit without consensus on the proper closure assumptions. Still, the computational effort for the numerical solution of the population balance embedded in the multizonal model is extensive, and it is difficult to extend this approach to multiple state variables necessary for dynamic metabolic models. This is an important argument to favor the alternative method of an agent-based Lagrange-Euler approach discussed in Section 3.5. 

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